m 


Student  Wohk— State  College  of  Washington. 


SHADES  AND  SHADOWS 


BY 

DAVID  C.  LANGE,  M.S. 

INSTRUCTOR  IN  ARCHITECTURE  IN  THE  WASHINGTON 
STATE  COLLEGE 


NEW  YORK 

JOHN   WILEY   &   SONS,   Inc. 

London:  CHAPMAN  &  HALL,  Limited 
1921 


:_3 


Copyright,  1921 

BY 

DAVID   C.  LANGE 


•    PRESS   OF  «■ 

eRAUNWORTH    &   CO. 

eOOK  MANUFACTURERS 

BROOKLYN,   N.   Y. 


PREFACE 


It  has  been  the  endeavor  of  the  author  to  give  in  the 
following  pages  sufficient  information  to  enable  one  to 
cast  correctly  the  Shades  and  Shadows  on  any  architectural 
object. 

The  information  contained  herem  was  compiled  with 
special  attention  for  its  use  as  a  text  book  on  Shades  and 
Shadows.  Architectural  students  in  many  colleges  receive 
their  early  training  under  Engineering  teachers.  An  at- 
tempt Vv'as  therefore  made  to  serve  such  students,  by 
assuming  the  point  of  view  of  their  engineering  training 
at  the  beginning  of  the  book,  and  lead  them  by  a  study  of 
Shades  and  Shadows  toward  an  appreciation  of  the  artistic 
architectural  point  of  view,  so  seldom  developed  in  a  strictly 
engineering  course. 

The  author  wishes  to  acknowledge  his  indebtedness  to 
the  Faculty  of  the  Architectural  School  of  the  University 
of  Pennsylvania,  where  he  received  his  first  impressions  and 
appreciation  of  Architectm-e,  and  especially  to  Professor 
Thomas  Nolan  for  his  able  assistance  in  a  review  of  the 
subject  matter  contained  in  the  book.  Also  to  those  whose 
work  is  shown  as  illustrations. 


4G8462 


CONTENTS 


Chapter  Page 

I.  Elementary  Prinxiples  of  Descriptive  Geometry. 

Points,  Lines,  and  Planes 1 

Intersections  of  Solids  with  Planes 27 

Intersection  of  Solids 34 

II.  Principles  of  Shades  and  Shadows. 

General  Principles  of  Shades  and  Shadows 48 

Shadows  of  Points 58 

Shadows  of  Lines G6 

Shadows  of  Planes 78 

Shades  and  Shadows  of  Solids 82 

Genera!  Methods  of  Finding  Shades  and  Shadows 90 

Wash  Rendering 130 


NOTE 


Problems  are  referred  to  as  plates. 

No  illustrations  accompany  the  following: 

Plates  I  to  XIII  Pages  36  to  44 

Plates  I  and  II  Pages  63  and  64 

Plates  III,  IV,  V  Pages  75  and  76 

Plates  XXVII,  XXVIII,  XXIX  Page  135 


SHADES  AND  SHADOWS 


CHAPTER  I 

ELEMENTARY  PRINCIPLES  OF  DESCRIPTIVE  GEOMETRY 
POINTS,    LINES    AND    PLANES 

1.  Descriptive  Geometry  is  the  art  of  representing 
a  body  in  space  upon  two  planes  (called  the  Horizontal,  H, 
and  the  Vertical,  V,  Coordinates,  indefinite  in  extent  and 
intersecting  at  right  angles  to  each  other  in  a  line  called 
the  Ground  Line),  by  projecting  lines,  perpendicular  to  the 
coordinates,  from  points  of  intersection  of  the  contiguous 
sides  of  the  body  and  from  points  of  its  contour,  and  thus 
solving  graphically  many  geometrical  problems  involving 
three  dimensions.     (Figs.  1,  2,  and  3.) 

NOTATION 

The'f oUowing  notation  is  used : 

H  Horizontal  Coordinate  Plane, 

V  Vertical  Coordinate  Plane, 

P  Profile  Coordinate  Plane, 

Q,  P,  R,  S,  etc.,  any  other  planes. 

HQ,  HP,  etc..  Horizontal  Traces  of  any  other  planes, 

VQ,  VP,  etc..  Vertical  Traces  of  any  other  planes, 

a,  b,  c,  d,  etc.,  any  points  in  space, 

a",  6",  etc.,  Vertical  Projections  of  any  points. 


PRINCIPLES  OF   DESCRIPTIVE   GEOMETRY 


2nJ  Qua  Irant 


Vertical  Coordinate 

Horizontal  Coordinate 


16"      \ch 


Fig.  3 


Ii«.  4 


G 


Fig.  5 


POINTS,   LINES  AND   PLANES  3 

a!",  h^,  etc.,  Horizontal  Projections  of  any  points, 

a^,  b'',  etc.,  Profile- Projections  of  any  points, 
A,  B,  C,  D,  etc.,  any  lines  in  space, 

A",  B",  etc..  Horizontal  Projections  of  any  lines, 

A'',  B' ,  etc..  Vertical  Projections  of  any  lines, 

A^',  5",  etc.,  Profile  Projections  of  any  lines, 

GL,  .  Ground-Line. 

2.  The  intersection  of  these  Coordinate  Planes  form 
four  angles  or  quadrants,  called  the  First,  Second,  Third, 
and  Fourth  Angles,  or  Quadrants.     (Fig.  1.) 

3.  In  order  to  represent  the  two  projections  of  an  object 
on  the  same  sheet  of  paper,  the  upper  portion  of  the  Ver- 
tical Plane  is  revolved  backward  about  the  Ground-Line 
as  an  axis,  until  it  coincides  with  the  Horizontal  Plane. 
(Fig.  2.) 

4.  All  points  in  the  First  Quadrant  are  vertically  and 
horizontally  projected,  respectively  above  and  below  the 
Ground-Line.  All  points  in  the  Third  Quadrant  are 
horizontally  and  vertically  projected,  respectively  above 
and  below  the  Ground-Line.  All  points  in  the  Second 
Quadrant  are  projected  above  the  Ground-Line.  All 
points  in  the  Fourth  Quadrant  are  projected  below  the 

^  Ground-Line.     (Figs.  2  and  3.) 

5.  Two  projections  of  a  point  are  always  on  one  and  the 
same  straight  line,  perpendicular  to  the  Ground-Line. 
(Fig.  3.) 

6.  Two  projections  are  always  necessary  definitely  to 
locate  a  point  or  line  in  space. 

7.  The  distance  of  a  point  in  space  above  the  Horizontal 
Plane  is  equal  to  the  distance  of  the  Vertical  Projection  of 
the  point  from  the  Ground-Line;  and  the  distance  of  a 
point  in  space,  in  front  of  the  Vertical  Plane,  is  equal  to  the 


PRINCIPLES   OF   DESCRIPTIVE   GEOMETRY 


.vjr 


I// 

Fig.  6 


V  .Fig.  7 


1      1                             II 

j   1    ^^    Jrf" 

Fig.  9 


e\ ,f" 

I  Jc'' 


Fig.  10  Fig.  11 


Fig.  12  Fig.  13 


Fig.  14 


POINTS,   LINES   AND   PLANES  5 

distance  of  the  Horizontal  Projection  of  the  point  from  the 
Ground-Line.     (Figs.  2  and  3.) 

8.  A  point  situated  on  either  Coordinate  Plane  is  its 
own  projection  on  that  Plane,  and  its  other  projection  is  in 
the  Ground-Line.     (Fig.  6.) 

9.  A  straight  line  is  determined  by  the  points  at  its 
extremities,  and  a  solid  is  made  up  of  lines;  hence  the  pro- 
jections of  a  sufficient  number  of  points  of  a  line  or  solid 
determine  their  entire  projections.     (Fig.  7.) 

10.  Lines  parallel  in  space  have  their  projections  par- 
allel.    (Fig.  8.) 

11.  A  straight  line  perpendicular  to  either  Coordinate 
Plane  has  its  projection  on  that  plane  as  a  point,  and  its 
other  projection  is  perpendicular  to  the  Ground-Line. 
(Fig.  9.) 

12.  A  line  parallel  to  either  plane  has  its  projection 
on  that  plane,  parallel  to  itself,  and  equal  to  its  true  length; 
and  its  other  projection  is  parallel  to  the  Ground-Line. 
(Figs.  10  and  11.) 

13.  A  line  parallel  to  both  coordinates,  or  to  the  Ground- 
Line,  has  both  projections  parallel  to  the  Ground-Line. 
(Fig.  12.) 

14.  A  point  in  a  line  has  its  projections  on  the  pro- 
jections of  the  line.     (Fig.  13.) 

15.  Projecting  Planes  are  planes  containing  two  or 
more  projecting  lines. 

16.  The  Traces  of  a  line  or  plane  are  the  intersections  of 
the  line  or  the  plane  with  the  coordinates,  and  the  two  traces 
of  the  same  plane  must  always  meet  in  the  same  point  in 
the  Ground-Line,  although  sometimes  at  infinity,  when 
the  plane  is  parallel  to  a  coordinate.  The  traces  of  a  line 
lying  in  a  plane  must  also  be  in  the  traces  of  that  plane. 
(Figs.  14  and  15.) 


6  PRi\riPLP:s  OF  descriptive  ceometry 


POINTS,   LINES  AND   PLANES  7 

17.  The  Profile  Plane  is  a  plane  perpendicular  to  the 
Ground-Line  and  hence  to  the  Horizontal  and  A'ertical 
Coordinates,  taken  for  convenience  on  the  left  side.  Pro- 
jections of  objects  on  the  Profile  Plane  are  shown  by  revolv- 
ing the  Profile  Plane  about  its  Vertical  Trace  into  the 
Vertical  Plane. 

18.  If  a  straight  line  is  perpendicular  to  a  plane,  its 
projectians  are  respectively  perpendicular  to  the  traces  of 
the  plane.  For  if  an  assumed  plane  is  perpendicular  to 
the  given  plane  and  a  Coordinate  Plane,  and  contains  the 
straight  line,  it  is  perpendicular  to  their  line  of  intersection; 
and  any  line  in  this  plane  is  perpendicular  to  this  line  of 
intersection.  But  this  assumed  plane  is  also  a  Projecting 
Plane  of  the  given  line,  and  the  lines  of  projection  must 
lie  in  the  traces  of  this  assumed  plane.  Projections  of  the 
lines,  therefore,  are  respectively  perpendicular  to  the  traces 
of  the  plane.     (Fig.  16.) 

19.  To  find  the  traces  of  a  given  line. 

Let  A  be  the  given  line.     (Figs.  17  and  18.) 

The  traces  of  a  line  are  the  points  in  which  the  line 
pierce^  the  Coordinate  Planes.     (Art.  16.) 

The  projections  of  these  traces  must,  therefore,  lie  in 
the  projections  of  the  line,  and  one  projection  oi  each  trace 
^lies  in  the  GL.  (Arts.  14  and  8.)  The  other  projection 
lies  at  the  intersection  of  a  line  perpendicular  to  the  GL 
through  this  point  in  the  GL,  and  the  projection  of  the  line. 
(Art.  6.) 

To  obtain  the  H  Trace  of  the  line,  the  line  is  continued 
to  the  H  Plane,  which  is  shown  by  the  Vertical  Projection 
intersecting  the  GL.  This  is  the  V  Projection  of  the  H 
Trace;  and  the  H  Projection  is  the  intersection  of  a  per- 
pendicular to  the  GL  at  this  point  with  the  H  Projection 
of  the  line.     The  V  Trace  is  found  in  the  same  way. 


8  PHIXCIPLE!^  OF   DESCRIPTIVE   GEOMETRY 


Fig.  22 


POINTS,    LINES   AND   PLANES  9 

20.  To  determine  the  true  length  of  a  hne  joining  two 
points  in  space. 

Let  A  be  the  given  Hne.     (Figs.  19  and  20.) 
A  line  is  seen  in  its  true  length  upon  that  coordinate 
to  which  it  is  parallel  or  in  which  it  lies.     By  revolving 
the  line,  therefore,  into  or  parallel  to  either  coordinate, 
the  true  length  of  the  line  in  space  is  determined.     This 
is  done  in  two  ways:    (1)  By  revolving  the  line  parallel 
to  a  Coordinate  Plane,  by  means  of  revolving  the  Pro- 
jecting Plane  parallel  to  the  opposite  coordinate.     (2)  By 
i    revolving  the  Projecting  Plane  about  its  intersection  with 
i    the  same  Coordinate  Plane,  as  an  axis,  into  that  coordinate. 
!  21.  To  pass  a  plane  through  two  intersecting  or  par- 

allel lines. 

Let  A  and  B  be  two  intersecting  lines.  (Figs.  21  and 
22.) 

The  traces  of  a  plane  must  contain  the  traces  of  the 

lines.    The  traces  of  the  given  lines,  therefore,  are  deter- 

1    mined,  and  like  traces  connected.     These  lines  are  the  traces 

i    of  the  required  plane,  and  meet  in  the  GL.     (Arts.  16  and 

19.) 

A  plane  may  be  passed  through  any  three  points  by 
!^  passing  two  intersecting  lines  through  the  three  points. 

22.  Given  one  projection  of  a  line  or  point  lying  in  a 
plane,  to  find  the  other  projection. 

'  Let  a^b''  be  the  H  Projection  of  the  line,  and  c^  be  the 
;  H  Projection  of  the  point,  lying  in  the  plane  P.  (Figs. 
I  23  and  24.) 

The  traces  of  the  line  in  the  given  plane  containing  the 
given  point  must  lie  in  the  traces  of  the  plane.  The  pro- 
jections of  the  traces  are,  therefore,  determined,  and  from 
them  the  unknown  projection.     (Arts.  16  and  19.) 

23.  Given  the  projections  of  a  point  or  line  lying  in  a 


10 


l'inX(II'Li:S   OF    DESCRIPTIVE    (lEOMETRY 


POINTS,    LINES   AND    PLANES  11 

plane,  tc  find  its  j)()8iti()n  when  the  plane  is  revolved  al)()ut 
its  trace  to  coincide  with  either  coordinate. 

Let  ab  be  the  line  and  c  the  point,  lying  in  the  plane 
P,  and  shown  by  their  projections.     (Fig.  25.) 

The  axis  about  which  the  point  or  points  of  the  line 
revolve  must  lie  in  the  plane  into  which  the  point  or  points 
of  the  line  is  revolved.  The  revolving  points  describe  a 
circle  whose  plane  is  perpendicular  to  the  line  of  inter- 
section of  the  given  plane  with  the  Coordinate  Plane.  The 
intersections  of  this  circle  or  circles  with  the  coordinate 
are  the  required  positions  of  the  point  or  points  in  the  line. 

Through  the  given  point  on  the  plane  a  plane  is  passed, 
perpendicular  to  the  trace  of  the  given  plane,  about  which 
trace  the  given  plane  is  to  be  revolved.  The  trace  of  the 
Auxiliary  Plane  is  perpendicular  to  the  axis  of  the  revolving 
plane.  On  the  trace  of  the  Auxiliary  Plane  a  point  is  laid 
ofT  at  a  distance  from  the  axis  and  trace  of  the  given  plane 
equal  to  the  hypotenuse  of  a  right  triangle,  one  side  of 
which  is  the  distance  from  the  projection  of  the  point  to 
the  axis  on  the  revolving  plane,  and  the  other  side  of  which  is 
equal  to  the  distance  of  the  point  from  the  same  projection. 

24.  To  find  the  true  size  of  an  angle  made  by  two  inter- 
secting lines.     (Fig.  26.) 

Let  A  and  B  be  the  tw^o  intersecting  lines. 

The  angle  betv/een  intersecting  lines  may  be  measured 
when  a  plane  containing  the  lines  has  been  revolved  to 
coincide  with  one  of  the  Coordinate  Planes.  A  plane  is, 
therefore,  passed  through  the  two  intersecting  lines  and 
its  traces  determined.  This  plane,  with  its  lines,  is  then 
revolved  about  either  of  its  traces  as  an  axis,  until  it  coin- 
cides with  that  Coordinate  Plane  in  which  the  trace  lies. 
The  angle  made  by  the  intersecting  lines,  in  the  revolved 
position,  is  the  required  angle.     (Art.  23.) 


12 


PRINCIPLES   OF    DESCRIPTIVE   GEOMETRY 


POINTS,   LINES  AND   PLANES  13 

25.  To  find  the  true  size  and  shape  of  any  plane  surface. 
Let  abed  be  the  plane  surface.     (Fig.  27.) 

This  plane  surface  appears  in  its  true  size  and  shape 
^vhen  the  plane  containing  it  is  revolved  to  coincide  with 
a  Coordinate  Plane.  The  plane,  therefore,  containing  the 
given  plane  surface,  is  revolved  into  the  coordinate  plane 
about  one  of  its  traces  as  an  axis,  and  the  revolved  position 
of  the  plane  surface  constructed.  This  is  the  true  size 
and  shape  of  the  plane  surface.     (Arts.  23  and  24.) 

26.  To  find  the  shortest  distance  from  a  point  to  a  line, 
and  to  draw  the  projections  of  it  in  its  position  in  space. 

Let  A  be  the  line  and  a  the  point.     (Fig.  28.) 

The  shortest  distance  from  a  point  to  the  line  is  a  per- 
pendicular from  the  point  to  the  line.  A  plane,  therefore, 
is  passed  through  the  line  and  point.  (Art.  2L)  This 
plane,  containing  the  line  and  point,  is  revolved  about  its 
trace  into  the  Coordinate  Plane.  This  determines  the 
revolved  position  of  the  line  and  point.  (Art.  23.)  A 
perpendicular  line  is  now  drawn  from  the  revolved  position 
of  the  given  point  to  the  revolved  position  of  the  given 
line.  This  is  the  true  length.  This  perpendicular,  revolved 
back,  determines  its  projections. 
(r       27.  To  find  the  line  of  intersection  of  any  two  planes. 

Case  L  To  find  the  intersection  of  two  planes  inter- 
secting within  the  limits  of  the  drawing. 

Let  X  and  P  be  the  intersecting  planes.  (Figs.  29  to 
32.) 

The  intersection  of  the  Horizontal  Traces  must  be  a 
point  common  to  both  planes,  and  therefore  a  point  common 
to  their  line  of  intersection.  The  Vertical  Projection  of 
this  point  lies  in  the  GL.  Projections  of  another  point 
in  this  line  of  intersection  may  be  determined  in  the  same 
way  from  the  intersection  of  the  Vertical  Traces.     (Art.  8.) 


14  PRINCIPLES   OF    DESCRIPTIVE   GEOMETRY 


POINTS,    LINES   AND    PLANES 


15 


16  PRINCIPLES  OF  DESCRIPTIVE   GEOMETRY 

Having  given,  therefore,  the  projections  of  two  points 
in  the  hne  of  intersection,  the  projections  of  the  Une  itself 
are  easily  determined.     (Art.  9.) 

Case  II .  To  find  the  intersection  of  two  planes  when 
the  traces  do  not  intersect  within  the  limits  of  the  drawing. 
(Figs.  33  to  37.) 

A  series  of  Auxiliary  Planes  are  passed  parallel  to  a 
Coordinate  Plane.  These  Auxiliary  Planes  cut  from  the 
given  intersecting  planes  straight  lines  which  are  parallel 
to  the  Coordinate  Plane,  and  which  intersect  in  points 
common  to  both  the  given  planes,  and  lie,  therefore,  in 
their  line  of  intersection. 

The  following  is  another  proof  for  the  same  problem: 

The  line  of  intersection  between  the  two  planes  is  com- 
mon to  each  plane,  and  the  traces  of  the  line  of  intersection 
must,  therefore,  lie  in  the  traces  of  each  plane.  (Art.  16.) 
Hence  the  point  of  intersection  of  the  V  Traces  of  the  planes 
is  the  Vertical  Trace  of  the  required  line  of  intersection, 
with  its  H  Projection  in  the  GL.  Another  point  in  the 
line  of  intersection  is  the  intersection  of  the  H  Traces  of 
the  planes,  and  its  Vertical  Projection  is  in  the  GL.  Having 
then  the  two  projections  of  the  two  points  in  the  line  of 
intersection,  the  projections  of  the  line  of  intersection 
itself  is  easily  determined.     (Art.  6.) 

Case  III.  To  find  the  intersection  of  two  planes  when 
both  intersecting  planes  are  parallel  to  the  GL,  or  when 
one  of  the  intersecting  planes  contains  the  GL. 

Let  P  and  0  be  the  two  planes  parallel  to  the  GL.  (Figs. 
36  and  37.) 

On  the  Profile  Plane  the  traces  of  the  intersecting  planes 
parallel  to  the  GL  are  determined.  A  point  common  to 
both  traces  is  a  point  in  the  intersection  of  the  two  planes. 
Since  the  intersecting  planes  are  perpendicular  to  the  Pro- 


POINTS,    LINES   AND   PLANES 


17 


c) 

VP 

I 

vo 

A. 

A'' 

\ 

^\ 

HO 

Fig.  36 


18  PEINCIPLEvS   OF   DESCRIPTIVE   GEOiMETRY 


POINTS,    LINES   AND   PLANES  19 

tile  Plane,  so  also  must  the  line  of  intersection  be  perpen- 
dicular. The  V  and  H  Projections,  therefore,  of  this  line 
are  the  projections  of  the  line  of  intersection  of  the  two 
planes  parallel  to  the  GL. 

28.  To  find  where  a  line  pierces  a  plane. 

Let  P  be  the  given  Plane  and  A  the  given  line.  (Figs. 
3S  to  41.) 

The  given  line  must  intersect  the  given  plane  in  the 
line  in  which  any  Auxiliary  Plane  containing  the  given 
line  intersects  the  given  plane,  at  a  point  v>here  the  given 
line  crosses  the  line  of  intersection  of  the  Auxiliary  Plane 
and  the  given  plane. 

An  Auxiliary  Plane  is,  therefore,  passed  through  the 
line  to  intersect  the  given  plane.     (Art.  16.) 

The  Hne  of  intersection  is  then  determined  between 
the  given  plane  and  the  Auxiliary  Plane.     (Art.  27.) 

The  required  point  lies  on  this  line  of  intersection  and 
on  the  given  line,  or  at  the  intersection  of  these  two  lines. 

There  are  several  cases,  as  follows: 

Case  I.  WTien  any  Auxiliary  Plane  containing  the  line 
is  used.     (Fig.  39.) 

Case  II.  When  the  H  or  V  Projecting  Plane  is  used 
as  the  Auxiliary  Plane.     (Fig.  38.) 

Case  III.  ^Yhen  the  line  is  parallel  to  the  Profile  Plane, 
necessitating  the  use  of  the  Profile  Plane.     (Fig.  40.) 

Case  IV.  WTien  the  plane  is  defined  by  two  inter- 
secting lines.     (Fig.  41.) 

29.  To  find  the  shortest  distance  from  a  point  to  a  plane. 
Let  P  be  the  given  plane  and  a  the  given  point.     (Fig. 

42.) 

The  shortest  distance  must  be  measured  along  a  line 
from  the  point  perpendicular  to  the  given  plane,  and  the 
projections  of  this  line  are  perpendicular  to  the  traces  of 


20  I'RINCIPLKS   OF    DESCRIPTIVE   GEOMETRY 


POINTS,    LINES   AND    PLANES 


21 


22  1>K]N(1PLE.S   OF   DESCRIPTIVE   GEOMETRY 

the  given  plane.  (Art.  18.)  The  point  in  which  this  per- 
pendicuhir  j^erces  the  gi\'cn  plane  is  then  determined. 
(Art.  28.) 

To  find,  therefore,  the  length  of  the  perpendicular,  a 
projecting  plane  containing  the  perpendicular  line  is  revolved 
about  its  trace  into  the  coordinate,  where  its  true  length 
is  shown. 

30.  To  pass  a  plane  through  a  given  point  and  parallel 
to  a  g'ven  plane. 

Let  P  be  the  given  plane  and  a  the  given  point.  (Fig. 
43.) 

The  traces  of  the  required  plane  are  parallel  to  the  cor- 
responding traces  of  the  given  plane,  and  are  fully  known 
when  one  point  in  each  trace  of  the  required  plane  is 
determined. 

A  straight  line,  therefore,  through  the  given  point,  and 
parallel  to  either  trace  of  the  given  plane,  is  a  line  in  the 
required  plane,  and  intersects  a  Coordinate  Plane  in  a  point 
in  the  trace  of  the  required  plane.  Thus  the  required 
plane  has  its  traces  through  this  point  and  parallel  to  the 
traces  of  the  given  plane.     (Art.  10.) 

31.  To  pass  a  plane  through  a  given  point,  perpen- 
dicular to  a  given  line. 

Let  a  be  the  given  point  and  A  the  given  line.  (Fig. 
44.) 

The  V  and  H  Traces  of  the  required  plane  are  per- 
pendicular to  the  V  and  H  Projections  of  the  given  line. 
(Art.  18.) 

The  direction  of  each  of  the  required  traces  is,  there- 
fore, known;  and  if  a  straight  line  is  drawn  through  the 
point,  and  ])aral!el  to  either  of  these  traces,  it  is  a  line  of 
the  reciuired  plane.  Unless  parallel  to  the  GL,  it  inter- 
sects one  of  th(>  i)lanes  of  projection  at  a  point  in  the  trace 


24  PRINCIPLES  OF  DESCRIPTIVE   GEOMETRY 

of  the  required  plane.  (Art.  16.)  Therefore,  a  trace 
through  the  point  thus  found,  and  perpendicular  to  the 
corresponding  projection  of  the  given  line,  is  one  of  the 
required  traces  of  the  required  plane.  The  other  trace 
meets  it  in  the  GL,  and  is  perpendicular  to  the  other  pro- 
jection of  the  line.     (Art.  16.) 

32.  To  pass  a  plane  through  a  given  line,  parallel  to 
another  given  line. 

Let  A  and  B  be  the  given  lines.     (Fig.  45.) 
The  required  plane   contains  one  of   the   given  lines, 
and  a  line  intersecting  this  given  line  which  is  parallel  to 
the  second  given  line. 

Through  any  point,  therefore,  in  the  first  line,  a  line 
is  passed  parallel  to  the  second  given  line.  A  plane  con- 
taining these  intersecting  lines  is  the  required  plane.  (Art. 
21.) 

33.  To  pass  a  plane  through  a  given  point  parallel  to 
two  given  lines. 

Let  A  and  B  be  the  given  lines,  and  a  the  given  point. 
(Fig.  46.) 

If  through  the  given  point  lines  are  passed  parallel 
respectively  to  the  given  lines,  and  a  plane  passed  through 
these  intersecting  lines,  this  plane  is  the  required  plane. 
(Art.  31.) 

34.  To  pass  a  plane  through  a  given  line,  perpendicular 
to  a  given  plane. 

Let  A  be  the  given  line  and  P  the  given  plane.    (Fig.  47.) 
The  required  plane  Q,  contains  two  intersecting  lines, 
namely,  the  given  line,  and  an  intersecting  line  B,  per- 
pendicular to  the  given  plane.     (Arts.  18,  21,  and  19.) 

35.  To  construct  the  projections  of  the  shortest  line  that 
can  be  drawn,  terminating  in  two  straight  lines  not  in  the 
same  plane. 


POINTS.    LINES   AND    PLANES 


25 


26 


PRINCIPLES   OF    DESC^'UPTIVE   GEOMETRY 


is  the  required  angle  which  can  be 
seen  in  its  true  size  by  revolving  it 
and  the  plane   O  containing  it  about 
its  traces  into  either  coordinate. 


INTERSECTIONS  OF  SOLIDS  WITH  PLANES  27 

Let  A  and  B  be  two  straight  lines  not  in  the  same  plane. 
(Fig.  48.) 

The  shortest  distance  between  two  points  not  in  the 
same  plane  is  the  perpendicular  distance  between  them, 
and  only  one  perpendicular  can  be  drawn  terminating  in 
these  two  lines. 

Through  one  of  the  given  lines  a  plane  is  passed  parallel 
to  the  second  line.  (Art.  32.)  The  second  given  line  is 
projected  on  this  plane.  (Arts.  12  and  29.)  This  pro- 
jection of  the  second  given  line  on  the  Auxiliary  Plane, 
intersects  the  first  given  line. 

At  their  intersection  a  line  E,  perpendicular  to  the 
Auxiliary  Plane,  is  dra\\Ti.  (Art.  18.)  It  intersects  the 
second  given  line  because  it  is  a  Projecting  Line,  and  is 
the  shortest  distance  between  the  lines.     (Art.  26.) 

36.  To  find  the  angle  made  by  any  two  intersecting 
planes. 

Let  P  and  R  be  two  intersecting  planes.  (Fig.  49.) 
A  plane  is  passed  which  is  perpendicular  to  the  line  of 
intersection  of  the  two  planes.  It  is  perpendicular  to  both 
'of  the  intersecting  planes,  and  cuts  from  each  a  line  per- 
pendicular to  the  line  of  intersection.  (Art.  18.)  The 
I  angle  made  by  these  lines  cut  out  by  the  perpendicular 
i;^lane  is  the  required  angle,  and  can  be  seen  in  its  true  size 
Iby  revolving  the  plane  containing  it,  about  its  trace,  into 
jpne  of  the  Coordinates.     (Art.  24.) 

INTERSECTIONS   OF   SOLIDS   WITH    PLANES 

36.  A  solid  is  a  magnitude  that  has  length,  breadth,  and 
|thickness,  as  a  Cylinder,  Cone,  or  Sphere. 

A  Cylinder  is  a  solid  generated  by  a  straight  line  called 
the  Generatrix,  moving  with  all  its  positions  parallel  along 
curved  lines  called  the  Directrix,  the  two  curved  lines  lie 


28 


PRINCIPLES  OF   DESCRIPTIVE   GEOMETRY 


d"       «"<•"      b" 


Fig.  50 


ab  is  an  element 


Fig.  52 


Fig.  53 


INTERSECTIONS  OF  SOLIDS  WITH  PLANES  29 

in  planes  forming  the  remainder  of  the  boundary.  The 
different  positions  of  the  Generatrix  are  called  the  Elements. 
(Fig.  50.) 

A  Right  Cylinder  is  one  whose  bases  are  parallel  planes 
perpendicular  to  its  axis;  and  a  Right  Section  is  a  circular 
section  cut  out  by  a  plane  perpendicular  to  its  axis.    (Fig.  51 .) 

An  Oblique  Cylinder  is  one  whose  bases  are  parallel 
planes  which  are  not  perpendicular  to  its  axis. 

A  Cone  is  a  solid  generated  in  a  manner  similar  to  that 
of  a  Cylinder,  except  that  its  elements  pass  through  a  fixed 
point  called  the  Vertex.     (Fig.  52.) 

A  Right  Cone  is  one  whose  base  is  a  plane,  perpendicular 
to  the  axis  of  the  Cone.     (Fig.  53.) 

A  Polyhedron  is  a  solid  bounded  by  plane  surfaces. 
(Fig.  54.) 

The  surface  of  a  Cylinder  or  Cone  are  called  single- 
i  curved  surfaces  of  revolution. 

I  The  surfaces  of  a  Sphere,  Ellipsoid,  Torus,  etc.,  arc 
'  called  double-curved  surfaces  of  revolution.  They  are 
generated  by  a  curved  line  revolving  about  another  line, 
called  an  Axis.  The  generating  curved  line  is  called  the 
Meridian  Line  and  the  curved  surface  it  generates,  a  Merid- 
:^an  Plane. 

38.  To  find  the  intersection  of  a  solid  with  a  Secant 
Plane. 

Let  P  be  the  Secant  Plane  intersecting  different  solids. 
(Figs.  54  to  57.) 

Auxiliary  Planes  are  passed  through  the  solid  and  the 
given  plane.  They  cut  straight  lines  from  the  plane  and 
straight  or  curved  lines  from  the  solid.  The  intersection 
of  one  of  these  lines  cut  from  the  plane  with  the  corre- 
sponding line  cut  from  the  solid  by  an  Auxiliary  Plane  is 
a  point  of  the  required  line  of  intersection. 


30  PRINCIPLES   OF    DESCRIPTIVE   GEOMETRY 


INTERSECTIONS  OF  SOLIDS  WITH    PLANES 


31 


32  PRINCIPLES   OF   DESCRIPTIVE   GEOMETRY 

While  the  AuxiHary  Planes  may  be  taken  in  any  position, 
yet  for  simplicity  they  should  be  chosen  so  as  to  cut  the 
simplest  line  or  curve  from  the  solid.  In  the  case  of  a 
solid  having  rectilinear  elements,  as  a  Cylinder,  Cone, 
Prism,  Pyramid,  etc.,  the  proceeding  described  is  prac- 
tically the  same  as  determining  where  a  certain  number  of 
elements  or  edges  pierce  the  Secant  Plane,  since  the  inter- 
section of  the  Auxiliary  Plane  by  these  elements  determines 
the  lines  cut  out  from  the  solid.  (Fig.  54.)  The  true  size 
of  the  section  cut  from  the  solid  can  always  be  found  by 
revolving  it  about  the  trace  of  the  Secant  Plane  into,  or 
parallel  to,  a  Coordinate  Plane.     (Art.  25.) 

39.  To  find  the  intersection  of  a  Cylinder  with  an 
oblique  plane. 

Let  a  Cylinder  be  cut  by  the  oblique  plane  P.  (Fig. 
55.)  Auxiliary  Planes  are  assumed  which  cut  elements 
from  the  Cylinder.  These  plane 3  are  parallel  to  the  axis 
of  the  Cylinder,  and  in  order  to  cut  the  simplest  lines  they 
should  be  perpendicular  to  one  of  the  coordinates,  as  U, 
T,  S,  R,  etc.  Each  Auxiliary  Plane  cuts  two  elements 
from  the  Cylinder,  and  a  straight  line  from  the  Secant 
Plane.  The  intersections  of  these  elements  with  the 
straight  line  cut  from  the  Secant  Plane  are  two  points  on 
the  required  curve  of  intersection,  as  a,  b,  c,  d,  etc. 

40.  To  find  the  intersection  of  a  Cone  with  an  obHque 
plane. 

Let  a  Cone  be  cut  by  an  oblique  plane  P.  (Fig.  56.) 
Auxiliary  Planes  arc  passed  through  the  vertex  of  the  Cone 
and  p(M-pendicular  to  a  coordinate,  as  R,  S,  T,  etc.  These 
Auxiliary  Planes  cut  elements  from  the  Cone  which  inter- 
sect the  lines  cut  from  the  given  plane  by  the  Auxiliary 
Planes,  as  A,  B,  C,  D.  These  intersect  m  the  points 
a,  b,  c,  and  d  in  the  curve  of  intersection.     The  additional 


INTERSECTIONS  OF  SOLIDS  WITH    PLANES  33 


34  PRINCIPLES   OF   DESCRIPTIVE   GEOMETRY 

points  needed  to  determine  the  intersection  are  determined 
in  the  same  way. 

41.  To  determine  the  intersection  of  a  double-curved 
surface  of  revolution  with  a  plane. 

Let  the  double-curved  surface  of  revolution  be  cut  by 
the  plane  P.     (Fig.  57.) 

Auxiliary  Planes,  perpendicular  to  the  axis,  are  passed 
through  the  solid.  These  planes  cut  circles  from  the 
double-curved  surface  of  revolution,  and  straight  lines 
from  the  given  Plane,  as  R,  S,  X,  Y,  Z.  The  intersection 
of  these  will  locate  points  on  the  curve  of  intersection. 
Other  points  of  the  curve  of  intersection  are  determined 
in  the  same  way.  The  points  a,  h,  c,  d,  e,f,  g,  etc.,  determine 
the  intersection  of  the  double-curved  surface  of  revolu- 
tion with  the  given  plane. 

42.  To  find  the  intersection  of  a  Polyhedron  with  any 
oblique  plane. 

Let  the  Polyhedron  be  intersected  by  the  oblique  plane 
P.     (Fig.  54.) 

The  intersection  of  the  given  plane  with  each  of  the 
bounding  planes  of  the  polyhedron  is  found  (Art.  27),  and 
this  determines  the  required  intersection. 

INTERSECTIONS   OF   SOLIDS 

43.  To  find  the  intersection  of  any  two  solids.  Auxiliary 
Planes  are  passed  through  the  two  solids.  These  Auxiliary 
Planes  cut  lines,  either  straight  or  curved,  from  each  solid, 
and  the  intersection  of  these  lines  with  each  other  determine 
points  on  the  required  line  of  intersection.     (Fig.  58.) 

The  lines  cut  from  the  solids  by  the  Auxiliary  Planes 
are  C,  D,  etc.,  and  1,  2,  3,  4,  5,  etc.,  are  the  points  on  the 
required  line  of  intersection  which  are  determined  by  the 
intersection  of  these  lines. 


36  PRINCIPLES  OF   DESCRIPTIVE   GE(3METRY 

PROBLEMS 

All  problems  are  to  be  worked  out  with  accuracy  and 
the  draftsmanship  is  to  be  of  the  highest  quality,  as  the 
plates  are  exercises  in  drawing  as  well  as  solutions  of  prob- 
lems in  Shades  and  Shadows.  The  sheets  are  to  be  15  by 
22  inches  in  size  and  are  to  have  a  margin  f  inch  in  width. 
All  plates  are  to  be  numbered  at  the  middle  of  the  top  of 
the  sheet,  and  each  is  to  have  the  date  in  the  lower  left- 
hand  corner  and  the  name  in  the  lower  right-hand  corner. 
Special  attention  should  be  paid  to  the  lettering  and 
notation. 

PLATE  I 

Draw  the  projections  of  a  point  in  each  of  the  follow- 
ing positions: 

1.  In  the  First  Angle. 

2.  In  the  V  Plane,  between  the  First  and  Second  Quad- 
rants. 

3.  In  the  Second  Angle,  equally  distant  from  the  H 
and  V  Planes. 

4.  In  the  H  Plane,  betvveen  the  Second  and  Third 
Angles. 

5.  In  the  H  Plane,  in  front  of  the  V  Plane. 

6.  In  the  Third  Angle. 

7.  In  the  Fourth  Angle. 

8.  In  the  GL. 

9.  In  a  Profile  Plane. 

10.  In  the  First  Quadrant,  2  inches  from  H  and  1  inch 
from  V. 


PROBLEMS  37 

PLATE    II 

Draw  the  projections  of  a  point  in  each  of  the  follow- 
ing positions : 

1.  In  the  Third  Angle,   1   inch  from  H  and   l\  inch 
from  V. 

2.  In  T",  and  |  inch  below  H. 

3.  In  H,  and  2|  inches  back  of  V. 

4.  In  the  Fourth  Angle,  1|  inches  from  H  and  \\  inches 
from  V. 

5.  In  the  Second  Quadrant,   If  inches  from  H  and  2 
inches  from  V. 

6.  In  the  Profile  Plane,  equidistant  from  H  and  V. 

7.  In  the  Profile  Plane,  and  in  the  Second  Quadrant. 

8.  Find  the  projection  of  any  point  whose  projecting 
lines  pass  through  a  point  1  inch  from  the  GL. 

9.  In  the  T^  Plane,  and  1  inch  from  the  H  Plane. 

10.  In  the  GL  lies  one  projection  of  a  point.     The  point 
itself  is  1  inch  from  the  GL.     Locate  the  point  in  space. 

PLATE  III 

Draw  the  projections  of  a  line  in  each  of  the  following 
positions : 

1 .  In  the  First  Angle,  parallel  to  V  and  H. 

2.  In  the  Second  Angle,  parallel  to  T^  and  oblique  to  H. 

3.  In  H,  behind  V  and  inclined  towards  V. 

4.  In  the  Third  Angle,  perpendicular  to  H. 

5.  In  the  Third  Angle,  oblique  to  V  and  H. 

6.  In  the  First  Angle,  oblique  to  H  and  T^  and  in  a 
plane  perpendicular  to  the  GL. 

7.  In  a  plane  bisecting  the  Fourth  Angle. 

8.  Draw  the  projections  of  two  intersecting  lines  in  the 
Third  Angle. 


38  PRIXf'IPLES  OF   DESCRIPTIVE   GEOMETRY 

9.  Draw  the  projections  of  two  parallel  lines,  one  in  the 
First  Angle  and  one  in  the  Second  Angle. 

10.  Draw  the  projections  of  a  point  in  the  line  in  the 
sixth  position. 

PLATE  IV 

Show  the  three  projections  of  a  point: 

1.  Two  and  one-half  inches  to  the  right  of  P,  1  inch 
in  front  of  V ,  and  2  inches  above  H. 

2.  Two  inches  to  the  right  of  P,  2\  inches  behind  V, 
and  2  inches  above  H. 

3.  One-half  inch  from  P,  l\  inches  in  front  of  V,  and 
2  inches  below  H. 

4.  Three  inches  to  the  right  of  P,  2  inches  behind  V, 
and  in  H. 

5.  In  P,  H,  and  7. 

6.  In  the  first  Angle,  and  1  inch  from  P,  H,  and  V. 

7.  In  P,  and  equidistant  from  H  and  V. 

8.  In  the  GL,  and  2  inches  from  P. 

9.  In  P,  between  the  First  and  Second  Angles. 

10.  In  P  and  V,  and  1  inch  from  H. 

PLATE  V 

Find  the  projections  of  the  lines  in  the  following  posi- 
tions: , 

1.  Inclined  to  V  and  H,  in  the  Third  Angle. 

2.  Parallel  to  P,  inclined  to  H  and  V,  and  in  the  First 
Angle. 

3.  Inclined  to  V  and  H,  in  the  First  Angle. 

4.  Inclined  to  H,  and  parallel  to  V,  in  the  Third  Angle. 

5.  Perpendicular  to  V,  and  in  the  Third  Angle. 
0.  Parallel  to  the  GL,  in  the  Fourth  Angle. 


PROBLEMS  39 

7.  Inclined  to  H,  and  laying  in  V,  between  the  Third 
and  Fourth  Quadrants. 

8.  Intersecting  lines  in  the  Third  Angle,  one  parallel 
to  H,  and  inclined  to  V,  and  one  parallel  to  the  GL. 

PLATE  VI 

1.  Draw  the  projections  of  a  line  which  intersects  the 
GL  at  a  point  3  inches  to  the  right  of  P,  and  pierces  P  at 
a  point  If  inches  from  H  and  2\  inches  from  V. 

2.  The  point  a  is  2  inches  from  H,  1  inch  behind  V, 
and  \  inch  to  the  right  of  P.     The  point  6  is  2  inches  to 

'   the  left  of  P,  1  inch  in  front  of  V,  and  lies  in  H.     Draw 
the  projections  of  a  line  through  these  points. 

3.  Draw  the  projections  of  a  line  passing  through  the 
First,  Second,  and  Third  Quadrants. 

1         4.  Draw  two  lines  intersecting  in  the  Second  Quadrant, 
at  a  point  2  inches  from  H  and  V. 

5.  Draw  the  projection  of  any  line  passing  through  a 
point  in  H,  and  2  inches  behind  V.     Intersect  this  line  at 

1   this  point  with  a  line  passing  through  the  Second,  Third, 
aiid  Fourth  Angles. 

6.  In  any  oblique  line,  find  a  point  which  is  equidistant 
if  from  H  and  V. 

7.  The  point  a,  of  the  triangle  ohc,  is  3  inches  to  the 
■  right  of  P,  4  inches  above  H,  and  5  inches  in  front  of  V. 
'  The  point  6  is  7  inches  to  the  right  of  P,  3  inches  behind  F, 

and  4  inches  belo^v  H.     The  point  c  is  2  inches  below  H. 
\   Draw  the  projections  of  the  triangle  ahc. 

PLATE  Vn 

i         1.  Draw  the  projections  of  a  plane  which  is  perpen- 

'   dicular  to  H,  and  makes  an  angle  of  45°  with  V. 

'         2.  Given  a  plane  which  makes  an  angle  of  30°  with 


40  PRINCIPLES   OF   DESCRIPTIVE   GEOMETRY 

H,  and  whose  H  Trace  is  parallel  to  the  GL,  and  2  inches 
back  of  V .     Draw  the  traces. 

3.  Draw  the  traces  of  a  plane  parallel  to  the  plane, 
in  VII,  2. 

4.  Given  a  line  which  is  oblique  to  the  GL,  whose  H 
Trace  is  back  of  V,  and  whose  V  Trace  is  above  H.  Draw 
the  projections  of  a  line  in  this  position. 

5.  Given  a  plane  which  is  parallel  to  the  GL.  Draw 
the  projections  of  a  point  in  this  plane. 

6.  Draw  an  oblique  plane  and  show  its  traces  in  all 
quadrants. 

7.  Draw  a  plane,  perpendicular  to  H,  but  oblique  to 
the  other  planes  of  projection. 

8.  Draw  a  plane,  perpendicular  to  P,  but  oblique  to 
H  and  F,  and  passing  through  the  First  and  Fourth  Quad- 
rants. 

9.  Draw  the  traces  of  a  plane  perpendicular  to  P,  and 
parallel  to  F. 

10.  Draw  the  projection  of  lines  in  each  Quadrant,  and 
find  its  traces  in  the  H,  V,  and  P  planes. 

PLATE  VIII 

1.  Find  the  true  length  of  a  line  ah,  located  in  space 
as  follows:  One  end  of  the  line  a,  is  1  inch  from  H,  and 
\  inch  from  T'.  The  other  end  6  is  2  inches  from  H,  and  f 
inch  from  V.     The  line  itself  is  1^  inches  long. 

2.  Pass  a  plane  through  tv\'o  intersecting  lines  whose 
intersection  is  1  inch  from  H,  and  1^  inches  from  V. 

3.  Pass  a  plane  through  two  parallel  lines  as  follows: 
The  line  ah  has  a  point  a,  \  inch  from  V,  and  f  inch  from 
//.  The  point  6,  is  1  inch  from  H,  and  1  inch  from  V. 
The  line  cd  has  the  point  c,  f  inch  from  7,  and  1§  inches 


PROBLEMS  41 

from  H.  The  point  c?  is  1|  inches  from  H,  and  If  inches 
from  V. 

PLATE  IX 

1.  Given  a  plane  whose  traces  make  an  angle  of  45° 
with  the  GL,  and  the  H  Projection  of  a  point  which  lies 
in  the  plane  and  is  1  inch  from  the  H  Trace  of  the  given 
plane,  and  1  inch  from  the  GL.  Find  the  other  projection 
of  the  point,  and  show  its  position  when  it  is  revolved 
into  one  of  the  coordinates. 

2.  Consider  the  triangular  plane  surface  in  Plate  VI,  7, 
and  find  its  true  size  and  shape. 

3.  The  point  a,  in  space  is  2  inches  from  H  and  F, 
and  \  inch  from  P.     The  line  ch  has  the  point  c  at  one  end, 

1  inch  from  P,  and  §  inch  from  H  and  V.  Its  other  end 
6  is  2  inches  from  P,  and  3  inches  from  H  and  7,  and  it  is 
4  inches  long.  Find  the  shortest  distance  from  the  point 
to  the  line  and  draw  its  projections  in  space. 

4.  Two  planes  P  and  R  have  their  traces  3  inches  apart 
in  the  GL.  The  trace  of  P  make  an  angle  of  45°,  the  H 
Trace  of  R  an  angle  of  30°,  and  the  V  Trace  an  angle  of 
60°  with  the  GL.     Find  the  line  of  intersection  between 

^  the  two  planes  which  intersect. 

5.  The  traces  of  a  plane  S  are  parallel  to  the  GL,  and 

2  inches  from  it.  The  H  Trace  of  a  plane  T  is  parallel  to 
the  GL  and  1  inch  from  it.  Its  V  Trace  is  3  inches  from 
it  and  parallel  to  it.  Find  the  line  of  intersection  between 
the  t\To  planes. 

6.  The  traces  of  the  plane  P  make  an  angle  of  45° 
\vith  the  GL.  Consider  a  line  in  space  which  does  not 
intersect  the  given  plane,  and  determine  where  it  intersects 
the  plane. 

7.  A  point  a  is  3  inches  from  H,  21  inches  from  V,  and 


42  PRINCIPLES   OF    DESCRIPTIVE   GEOMETRY 

I  inch  from  P.  The  H  Trace  of  a  plane  makes  an  angle 
of  45°,  and  the  Y  Trace  an  angle  of  60°,  with  the  GL.  The 
point  of  intersection  of  the  traces  with  the  GL  is  |  inch  from 
P.  Find  the  shortest  distance  from  the  point  to  the  plane 
and  draw  its  projections. 

PLATE  X 

1.  The  point  a  is  2|  inches  from  P,  |  inch  from  H, 
and  f  inch  from  V.  Pass  a  plane  through  this  point  and 
parallel  to  another  plane  whose  traces  make  angles  of  60° 
and  30°,  respectively,  with  H  and  V. 

2.  In  the  preceding  problem  pass  a  plane  perpendicular 
to  the  given  plane. 

3.  A  point  a  is  2  inches  from  P,  1  inch  from  V,  and  f 
inch  from  H.  A  line  he  has  one  end  h,  J  inch  from  P, 
I  inch  from  V,  1  inch  from  H,  and  is  2  inches  long.  The 
other  end,  c  is  1  inch  from  V  and  1^  inches  from  H.  Another 
line  de  has  one  end  f  inch  from  P,  2  inches  from  V,  and 
1|  inches  from  H,  and  is  2|  inches  long.  The  other  end, 
e,  is  I  inch  from  V  and  H.  Pass  a  plane  through  the  point 
and  parallel  to  the  two  lines, 

4.  A  line  ab  has  its  end  a,  \  inch  from  P,  1  inch  from  V, 
and  H  inch  from  H.  Its  other  end,  h,  is  If  inches  from 
V,  and  2  inches  from  H.  The  line  is  2|  inches  long. 
Another  line  cd  has  its  end  c  f  inch  from  P,  3  inches  from 
H,  1  inch  from  V,  and  is  2|  inches  long.  Its  other  end 
d  is  1^  inches  from  V  and  If  inches  from  H.  Construct 
the  projection  of  the  shortest  line  that  can  ba  drawn  termi- 
nating in  these  two  given  lines. 

5.  The  traces  of  a  plane  R  make  an  angle  of  45°  with 
H  and  30°  with  7,  and  meet  in  a  point  in  the  GL,  \  inch 
from  P.  The  traces  of  another  plane  S  make  an  angle  of 
60°  with  //  and  45°  with  F,  and  meet  in  the  GL,  3  inches 


PROBLEMS  43 

from   P.     Find    the   true   size  of   the   angle  between   the 
planes. 

PLATE  XI 

1.  A  Right  Cylinder,  2  inches  in  diameter,  rests  on  H", 
and  has  its  4-inch  axis  3  inches  from  P  and  2\  inches  from 
V.  This  Cylinder  is  intersected  by  a  plane  whose  traces 
make  an  angle  of  30°  with  H  and  45°  with  V,  and  intersect 
in  a  point  4  inches  from  the  GL.  Determine  the  true  size 
of  the  section  cut  out. 

2.  Replace  the  Cylinder  in  the  above  problem  with  a 
Cone  of  the  same  size  and  determine  the  true  size  of  the 
section  cut  out. 

PLATE  XII 

j  1.  A  Sphere,  2|  inches  in  diameter,  has  its  center  1§ 
•  inches  from  H,  V,  and  P.  It  is  intersected  by  a  plane 
i  whose  traces  make  angles  of  30°  and  45°,  respectively,  with 
,  H  and  V,  and  meet  in  the  GL  at  a  point  2\  inches  from  P. 
Determine  the  line  of  intersection  cut  from  the  solid  by 
the  plane. 

2.  A  Cone  with  a  2-inch  base  rests  on  H,  with  its  center 
t;l|  inches  from  V  and  P.  Its  axis  is  3  inches  long  and 
makes  an  angle  of  45°  wdth  the  H  and  V  Planes,  sloping  up 
f(/rward,  to  the  right  {ufr).  Find  the  true  size  of  a  section 
cut  out  by  a  plane  making  with  its  H  Trace  an  angle  of 
45°,  and  with  its  V  Trace  an  angle  of  30°  with  the  GL, 
when  its  traces  meet  in  the  GL,  3|  inches  from  P. 

PLATE  XIII 

1.  Find  the  lines  of  intersection  between  two  solids, 
as  follows:  A  Cylinder,  2\  inches  in  diameter,  with  its 
base  resting  on  H,  its  center  4  inches  from  H  and  Ij  inches 


44  PRINCIPLES   OF   DESCRIPTIVE   GEOMETRY 

from  P,  has  a  5-inch  axis  which  slopes  up  back,  to  the 
right  (ubr).  A  cone,  with  a  3-inch  base  resting  on  H,  its 
center  4  inches  from  H  and  5  inches  from  P,  has  its  5-inch 
axis  sloping  up  back,  to  the  left  {uhl).  The  axis  of  the 
Cylinder  makes,  in  V  Projection,  an  angle  of  60°,  and  in 
H  Projection,  an  angle  of  30°  with  the  GL.  The  axis  of 
the  cone  makes,  in  V  Projection,  an  angle  of  45°,  and  in 
H  Projection,  an  angle  of  45°  with  the  GL. 


QUESTIONS   IN   DESCRIPTIVE   GEOMETRY  45 

QUESTIONS    IN    DESCRIPTIVE    GEOMETRY 

1.  Define  Descriptive  Geometry.  What  are  Coordinate 
Planes?  Projecting  Lines?  Projecting  Planes?  Quad- 
rants or  Angles? 

2.  What  is  done  in  order  that  two  projections  of  an 
object  may  be  shown  on  the  same  sheet  of  paper?  What 
is  the  Ground-Line?  What  angles  does  the  Horizontal 
Plane  make  with  the  Vertical  Plane? 

3.  How  are  all  points  in  the  First  Quadrant  projected? 
In  the  Second  Angle?     In  the  Third  and  Fourth  Quad- 

j  rants?     How    are    straight    lines    represented?     How    are 
I  their  projections  determined? 

4.  How  are  the  projections  of  a  point  shown  with  refer- 
ence to  the  Ground-Line?  How  many  projections  are 
necessary  to  locate  definitely  a  point  in  space?  Name  the 
different  projections  a  point  may  have. 

5.  What  does  the  distance  of  the  Vertical  Projection 
from  the  Ground-Line  show?  How  are  the  projections 
of  a  point  lying  in  one  of  the  Coordinates  shown? 

6.  When  the  projections  of  two  lines  are  parallel,  define 
the  position  of  the  lines  to  each  other  in  space?     If  a  straight 

^ine  is  perpendicular  to  either  Coordinate  Plane,  what  are 
the  positions  of  its  projections? 

7.  If  a  line  is  parallel  to  either  coordinate,  how  will 
its  projections  appear?  If  the  projections  of  a  line  are 
parallel  to  the  Ground-Line,  how  is  the  position  of  the  line 
in  space  determined  with  reference  to  the  Ground-Line? 

8.  If  a  point  is  on  a  line,  what  are  the  positions  of  its 
projections  with  reference  to  the  projections  of  the  line? 
Wliat  can  be  said  of  the  projections  of  the  intersection 
of  any  two  lines  in  space? 

9.  Can  the  intersection  which  a  line  makes  with  the 


46  PRINCIPLES  OF   DESCRIPTIVE   GEOMETRY 

Coordinate    Planes   determine  the   direction   of   a  line  in 
space? 

10.  Define  the  Traces  of  a  Plane.  Name  the  different 
traces  which  a  plane  may  have.  Where  do  the  traces  of 
any  plane  meet?     Explain  why  they  meet  where  they  do. 

1 1 .  Define  Profile-Plane.  Describe  the  traces  of  a  plane 
parallel  to  either  coordinate.  How  is  a  plane  containing 
the  Ground-Line  determined?  If  a  straight  line  is  per- 
pendicular to  a  plane,  what  are  the  positions  of  its  pro- 
jections in  relation  to  the  traces  of  the  plane? 

12.  Explain  the  notation  used  in  this  book.  If  a  plane 
is  parallel  to  the  Ground-Line,  what  are  the  positions  of 
its  traces?  If  only  one  trace  is  shown  on  the  Vertical 
and  Horizontal  Coordinates,  and  if  this  is  parallel  to  the 
GL,  what  is  the  position  of  the  plane? 

13.  if  a  plane  is  perpendicular  to  either  coordinate, 
what  are  the  positions  of  its  traces  on  the  opposite  coordi- 
nate. Where  do  the  traces  of  a  plane  stop?  Given  the 
traces  of  a  line,  how  are  its  projections  determined? 

14.  Explain  how  the  line  of  intersection  of  any  two 
planes  is  determined. 

15.  Make  the  drawings  and  give  the  proof  for  finding 
the  shortest  distance  from  a  given  point  to  a  line. 

16.  How  is  the  angle  which  a  given  line  makes  with 
any  oblique  plane  determined?  How  is  the  true  size  of  any 
plane  surface  determined? 

17.  Given  the  projections  of  a  point  or  line  lying  in 
a  plane,  determine  the  position  of  the  point  or  line  by  draw- 
ing and  explanation,  when  the  plane  is  revolved  to  coincide 
with  either  coordinate. 

18.  Make' the  drawing  and  give  the  explanation  for 
finding  the  true  size  of  an  angle  made  by  two  intersecting 
lines  in  space. 


QUESTIONS   IN    DESCRIPTIVE   GEOMETRY  47 

19.  Explain  the  relation  of  the  traces  of  a  line  lying  in 
a  plane  to  the  traces  of  the  plane.  Explain  the  method 
of  passing  a  plane  through  three  points  in  space.  How 
many  planes  can  be  passed  through  a  line  in  space? 

20.  Make  the  drawings  for  finding  the  angle  which  a 
given  line  makes  with  a  given  plane;  a  given  plane  with 
the  planes  of  projections? 

21.  Make  the  drawings  and  give  the  explanation  for 
finding  the  line  of  intersection  of  a  plane  with  a  polyhedron. 

22.  How  is  the  intersection  of  any  oblique  plane  with 
any  solid  determined?  Of  any  solid  with  any  other  solid? 
Give  the  full  explanation. 

23.  How  is  the. true  size  of  a  section,  cut  out  by  any 
oblique  plane  from  any  solid,  determined? 


CHAPTER  II 

I.     GENERAL  PRINCIPLES  OF  SHADES  AND  SHADOWS 

1.  Objects  are  visible  to  the  eye  because  of  the  reflec- 
tions from  their  surfaces  of  Rays  of  Light  of  varying  degrees 
of  intensity.  These  reflected  rays  impinge  on  the  retina 
of  the  eye,  and  result  in  visual  sensation  and  perception 
of  the  illuminated  object.  The  principal  source  of  light 
is  the  sun,  and  its  Rays  of  Light  fall  on  objects  at  various 
angles.  If  the  angle  of  incidence  of  all  the  Rays  imping- 
ing on  a  surface  are  the  same,  the  surface  appears  flat  and 
IS  a  plane  surface;  while  if  the  angles  of  incidence  are  dif- 
ferent, the  surface  is  not  a  plane  surface. 

2.  The  importance  of  the  subject  of  Shades  and  Shadows 
should  be  fully  realized,  especially  by  students  of  Archi- 
tecture. It  is  a  matter  of  surprise  that  few  students,  even 
after  finishmg  the  usual  course  of  study,  are  able  to  deter- 
mine accurately  the  Shades  and  Shadows  on  architectural 
drawings.  They  often  resort  instead  to  mere  copying  to 
the  use  of  Shades  and  Shadows  merely  guessed  at,  and  to 
a  lavish  rendering  to  cover  mistakes.  The  beauty  and 
impressiveness  of  a  drawing  is  more  truthfully  enhanced  by 
the  accurate  casting  of  its  shadows  than  by  its  rendering 
1  he  casting  of  shadows  on  architectural  drawings  is  a  part 
of  architecture  as  a  Fine  Art,  and  an  important  part  of  it« 
function  IS  to  give  a  detail  its  true  shape  and  show  its  true 
character. 

r^r^J^'-  ^T'"""""  "^  """"'""^  "'''^'^""S  *<'  funda- 
mental principles  cannot  be  impressed  to  ostrongly  upon 


GENERAL   PRINCIPLES 


49 


?^r! 


Fig.  1. — An  End  Pavilion 
Student  Work— Wm.  Hough.      University  of  Penna. 


50  PRINCIPLES  OP  SHADES  AND   SHADOWS 


IMHH  HHUBv  AktsMSss^r-:, 

m 

Jl||(^=^  >- '.'  ^^^^^^^K  '      ,. 

m  i 

4 

Fig  .2. 


GENERAL   PHLN'CIPLES  51 

the  student's  mind,  for  when  these  are  once  mastered  the 
rest  of  the  work  is  relatively  easy.  The  theory  may  be 
taught  in  a  very  short  time,  and  when  understood,  enables 
one  to  solve  easily  any  problem  in  the  subject.  It  is,  how- 
ever, necessary,  in  order  to  become  thoroughly  familiar 
with  the  character  of  the  Shades  and  Shadows  on  various 
common  architectural  forms  and  shapes,  that  certain  prob- 
lems be  accurately  solved. 

4.  The  subject  of  Shades  and  Shadows  is  an  appli- 
cation of  Descriptive  Geometry,  and  its  purpose  is  to  give 
a  more  realistic  appearance  to  the  representation  of  objects. 

5.  Architectural  drawings  may  be  divided  into  two 
general  classes.  Working  Drawings  and  Rendered  Draw- 
ings. It  is  by  the  use  of  Shades  and  Shadows  in  rendered 
drawings  that  a  truthful  and  realistic  representation  is 
made  of  an  object,  which  would  otherwise  appear  flat. 
All  elevations  and  plans  (Coordinate  Planes)  are  conven- 
tional, and  so  are  Shades  and  Shadows.  They  should  not 
be  made  to  produce  the  effect  of  perspective  when  the 
drawing  itself  is  not  in  perspective. 

6.  The  casting  of  •  Shades  and  Shadows  enables  one 
more  readily  to  determine  the  shapes  of  various  details. 
They  give  mass  and  proportion  to  objects,  and  with  their 
aid  a  more  intimate  understanding  of  architectural  com- 
positions is  possible.  Their  use  is  important,  both  in  the 
rendering  of  the  drawing  and  in  the  study  of  the  design. 
The  designer  models  his  building  with  the  aid  of  Shades 
and  Shadows,  and  gives  it  texture,  color,  relief,  and  pro- 
portion, so  that  cornices,  colonnades,  windows,  and  all 
architectural  details,  can  be  readily  comprehended  as  they 
appear  in  reality.     (Figs.  1  and  2.) 

7.  ShadoAvs  greatly  influence  design  and  the  character 
of  a  building.     This  becomes  evident  from  a  study  of  the 


52  PRINCIPLES   OF  SHADES   AND   SHADOWS 


iM.i.   3. 


Fig.  4. 


GENERAL  PRINCIPLES  53 

developments  of  architectural  styles  in  different  latitudes. 
Greek  architecture  with  its  simple  lines  could  not  have 
reached  its  perfection  in  northern  climates,  and  Gothic 
architecture,  which  was  developed  under  the  low-lying  sun, 
could  not  have  come  into  such  prominence  under  the  bril- 
liant southern  sky.     (Figs.  3  and  4.) 

8.  The  beauty  of  massing  and  proportion,  of  the  mold- 
ings and  other  details  of  a  building,  is  greatly  enhanced 
by  contrasts  of  light  and  shade. 

Profiles  of  moldings,  proportion,  and  massing,  are  of 
minor  importance  in  themselves,  if  not  considered  in  con- 
nection with  Shades  and  Shadows.     (Figs.  1  and  2.) 

9.  Since  the  sun  is  the  assumed  source  of  light,  con- 
sidered at  an  indefinite  distance  away,  all  its  Rays  of  Light 
are  parallel. 

10.  A  further  conventional  direction,  also,  for  the  Rays 
of  Light  which  has  been  agreed  upon,  is  the  average  of  the 
different  directions  of  the  sun's  rays.  This  Conventional 
Direction  for  the  Rays  of  Light  is  assumed  to  be  parallel  to 
the  diagonal  of  a  cube,  sloping  from  the  upper,  front,  left- 
hand  corner  to  the  lower,  back,  right-hand  corner,  when 
the  faces  of  the  cube  are  parallel  or  perpendicular  to  the 
vertical  or  horizontal  coordinates.     (Figs.  5  and  6.) 

11.  The  true  angle  \vhich  the  Conventional  Ray  of 
Light  makes  with  the  Coordinate  Planes  is  35°15'  22".  The 
projections  of  the  same  Ray  make  angles  of  45°  with  the 
GL,  or  Coordinate  Planes.     CFigs.  6  and  7.) 

127  A  Plane  of  Light  is  any  plane  containing  a  Ray 
of  Light.     (Fig.  8.) 

13.  Light  falling  upon  any  object  may  be  Direct, 
Indirect,  Diffused,  or  Artificial. 

Direct  Rays  of  Light  are  those  Rays  which  fall  directly 
on  any  object.     (Fig.  9.) 


54  PRINCIPLES   OF   SHADES   AND   SHADOWS 


Fig.  8 


GENERAL   PRINCIPLES  55 

Indirect  Rays  of  Light  are  those  Rays  which  are  reflected 
back  on  an  object  from  some  other  object.  This  is  called, 
also,  Reflected  Light.     (Fig.  1,  a  and  b.) 

Diffused  Light  is  that  light  which  is  widely  scattered, 
and  has  its  rays  spread  out  in  all  directions  from  the  sources 
of  light.     (Fig.  10,  page  129.) 

Artificial  Light  is  light  produced  by  artificial  illumi- 
nation as  the  source  of  light.     (Fig.  11,  page  129.) 

Direct  and  Indirect  Rays  of  Light  are  nearly  always 
used  in  rendering  architectural  drawings.  Diffused  Light 
is  often  used  in  rendering  interior  perspectives,  while  Arti- 
ficial Light  is  seldom  employed  in  rendering.  In  this 
Chapter  Direct  Rays  are  the  only  Rays  of  Light  considered. 

14.  Shade  is  that  portion  of  the  surface  of  a  body  which 
is  turned  away  from  the  source  of  light,  and  which  there- 
fore does  not  receive  any  of  the  Rays  of  Light.     (Fig.  9.) 

15.  Shadow  is  that  portion  of  any  surface  of  an  object 
from  which  light  is  obstructed  by  means  of  another  object 
placed  between  it  and  the  source  of  light.     (Fig.  9.) 

IG.  The  Shade-Line  is  the  boundary  line  of  the  Shade, 
and  the  Shadow  of  the  Shade-Line  of  any  object  always 
determines  and  bounds  the  Shadow  of  the  object  casting 
the  Shadow.  From  the  Shadow  of  an  object  its  Shade- 
Line  can  always  be  determined.     (Fig.  9.) 

17.  The  Umbra  is  that  portion  of  space  from  which 
light  is  excluded.  The  Umbra  of  a  point  is  a  line.  The 
Umbra  of  a  straight  line  is  a  straight  line  or  a  plane;  while 
the  Umbra  of  a  plane  is  a  plane  or  a  solid.     (Fig.  9.) 

18.  Real  Shades  and  Shadows  are  those  which  have 
actual  existence.  Imaginary  Shades  and  Shadows  are  those 
which  have  no  real  existence,  and  which  are  used  solely 
for  the  purpose  of  determining  the  real  Shades  and  Shadows. 
Invisible  Shades  and  Shadows  are  those  which  have  actual 


56  PRINCIPLES   OF  SHADES  AND   SHADOWS 


1 


Fie.  9 


Shadow  line 


Umbra 


Shade 

.Sliade  Line 


Invisible  shade 
Invisible  shade  line 


GENERAL  PRINCIPLES  57 

existence  but  cannot  be  seen  in  projection,  and  which  are 
therefore  dotted  in  on  a  drawing.     (Figs.  9  and  13.) 

19.  The  Shadow  of  any  object  is  found  by  passing 
Rays  of  Light  through  the  object,  and  determining  where 
the  Rays  intersecting  the  object  casting  the  Shadow  inter- 
sect the  surface  of  the  object  on  which  the  shadow  is  cast. 
(Fig.  9.) 

20.  The  subject  of  Shades  and  Shadows  of  objects  may 
be  conveniently  divided  into  (1)  Shadows  of  Points,  (2) 
Shadows  of  Lines,  (3)  Shadows  of  Solids,  and  (4)  Special 
Methods  for  Determining  Shades  and  Shadows,  c^ 

21.  Attention  is  again  called  to  the  fact  that  the  Y 
Projection  of  an  object  represents  the  elevation  of  the 
same  object,  the  H  Projection  the  plan,  and  the  P  Pro- 
jection the  side  elevation  or  section  of  the  same  object. 

22.  The  following  additional  notation  to  that  given 
in  Chapter  I  is  used  in  this  Chapter  II : 

ct%  b^%  etc.,  the  H  Projections  of  the  shadows  of  points  in 

space  on  any  surface, 
a",  h'%  etc.,  the  V  Projections  of  the  shadows  of  points  in 

space  on  any  surface. 
R,  a  Ray  of  Light. 

R'',  the  H  Projection  of  a  Ray  of  Light. 

/?",  the  V  Projection  of  a  Ray  of  Light. 

R^,  the  P  Projection  of  a  Ray  of  Light. 


58  SHADOWS  OF   POINTS 


n.     SHADOWS  OF  POINTS 


23.  To  find  the  shadow  of  any  given  point  in  space 
on  any  surface  or  plane. 

Let  a  be  any  point  in  space.     (Figs.  13  to  19.) 

The  point  in  which  a  Ray  of  Light,  passed  through  a 
point  in  space,  pierces  the  surface  receiving  the  shadow, 
is  the  shadow  of  the  point.  The  piercing-point  of  this 
Ray  of  Light  with  the  plane  receiving  the  shadow  is  the 
point  in  which  a  line  having  the  direction  of  the  Ray  of 
Light,  through  the  point,  pierces  the  plane  receiving  the 
shadow.     (Chap.  I,  Arts.  14  and  28.) 

Case  I.  When  the  shadow  of  a  given  point  falls  on  the 
Vertical  Coordinate,  or  is  shown  on  the  elevation  of  an 
object.     (Fig.  12.) 

Let  a  be  any  given  point,  shown  by  its  projections 
a*  and  a\     To  find  the  shadow  of  a. 

The  point  in  which  a  Ray  of  Light,  passed  through  the 
point  a  (shown  by  the  projections  of  the  Ray  of  Light, 
B!"  and  R%  through  the  projections  of  the  point,  a!"  and  a') 
pierces  the  V  Plane,  is  the  shadow  of  a  on  that  plane; 
and  it  is  graphically  indicated  by  the  point  in  which  a  per- 
pendicular, erected  at  the  intersection  of  the  H  Projection 
of  the  Ray  of  Light  with  the  GL,  intersects  the  V  Projection 
of  the  Ray  of  Light. 

Case  II.  When  the  shadow  of  a  given  point  falls  on 
the  Horizontal  Coordinate,  or  is  shown  on  the  plan  of  an 
object.     (Fig.  13.) 

This  case  is  the  same  as  the  preced  ng  one,  except 
that  the  Ray  of  Light  pierces  the  H  Plane  instead  of 
the  V  Plane,  and  therefore  the  H  Projection  of  the  shadow 
of  a  is  the  point  in  which  a  perpendicular  erected  at 
the  intersection  of  the  V  Projection  of  the  Ray  of  Light 


PRINCIPLES   OF  SHADES   AND   SHADOWS 


59 


Fig.  12 


Fig.  13 


Fig.  14 


Fig.  15 


60  SHADOWS  OF   POINTS 

with  the  GL,  intersects  the  H  Projection  of  the  Ray  of 
Light. 

Case  III.  When  the  shadow  of  any  point  falls  on  the 
GL,  or  on  both  coordinates.     (Fig.  14.) 

In  this  case  the  projections  of  the  Rays  of  Light  through 
the  projections  of  the  point,  intersect  the  GL  at  the  same 
point.  This  point  is  the  projections  of  the  shadow  of  the 
point. 

Case  IV,  To  find  the  Imaginary  Shadow  of  any  point 
behind  or  below  one  Coordinate  when  it  is  on  the  other 
Coordinate  Plane.     (Fig.  15.) 

This  is  the  problem  of  finding  the  point  in  which  the 
Ray  of  Light  through  the  given  point,  passed  through  the 
Coordinate  Plane  it  first  strikes,  as  though  this  plane  were 
transparent,  pierces  the  other  Coordinate  Plane,  behind  or 
below  the  first  Coordinate  Plane.  This  determines  the 
position  of  the  Imaginary  Shadow  of  the  point  on  that 
Coordinate  Plane. 

This  is  shown  graphically  by  continuing  that  projection 
of  the  Ray  of  Light  (which  first  intersects  the  GL)  through 
the  like  projection  of  the  point,  and  determining  the  point 
in  which  the  other  projection  of  the  Ray,  through  the  other 
projection  of  the  point,  intersects  the  GL.  The  intersec- 
tion of  a  perpendicular,  erected  at  the  last  mentioned  point, 
with  the  first  mentioned  projection  of  the  Ray  of  Light, 
is  the  required  projection  of  the  Imaginary  Shadow  of  the 
given  point. 

Case  V.  To  find  the  shadow  of  a  point,  having  given 
one  Coordinate  Projection,  either  plan  or  elevation,  and  the 
Profile  Projection.     (Figs.  16,  17,  and  18.) 

The  same  proof  applies  to  this  case  as  to  the  preceding 
cases.  The  graphical  method  of  finding  the  projections 
of  the  shadow  of  the  given  point  is  as  follows : 


i 


PHIXCIPLES   OF   SHADES   AND   SHADOWS 


()1 


Fig.  16 


^ \.a> 


Fig.  17 


Fig.  19 


. IQ" 


Fig.  18 


62  SHADOWS  OF   POINTS 


The  two  projections  of  the  Ray  of  Light,  R^  and  R 
or  R',  are  passed  through  the  projections  of  the  point. 
The  projection  of  the  shadow  of  the  point  is  the  point 
in  which  a  Hne,  erected  at  the  intersection  of  the  Profile 
Projection  of  the  Ray  with  the  H  or  V  Coordinate,  and 
perpendicular  to  the  Profile  Trace,  intersects  the  other 
projection  of  the  Ray,  or  intersects  a  perpendicular  erected 
at  the  point  in  which  the  other  projection  of  the  Ray  inter- 
sects the  GL. 

Case  VI .  When  the  shadow  of  a  given  point  falls  on 
any  oblique  surface  or  plane.     (Fig.  19.) 

Let  a  be  a  point  in  space  and  P  any  oblique  plane  on 
which  the  shadow  of  a  falls. 

This  problem  is  solved  in  the  same  way  as  the  preced- 
ing problems  in  Cases  I  to  V. 

The  graphical  solution  is  the  same  as  that  employed 
to  determine  the  point  of  intersection  of  a  line,  representing 
a  Ray  of  Light,  with  any  plane.     (Chap.  I,  Art.  28.) 

It  is  well  to  note  here  that  the  Ground-Line  can  be 
separated  into  two  Ground-Lines,  without  affecting  in  any 
way  the  resulting  shadows.  In  fact  these  are  the  usual 
conditions  with  which  one  has  to  contend  in  the  practical 
casting  of  Shades  and  Shadows  in  architectural  work, 
since  the  elevation  is  often  on  one  sheet  of  paper,  the  plan 
on  another,  and  the  section  on  a  third. 

Axiom  I.  A  point  in  a  plane  is  its  own  shadow  on  that 
plane. 


l\ 


PROBLEMS  63 


PROBLEMS 


The  same  note  applies  to  the  problems  in  the  following 
plates  as  to  the  problems  in  the  plates  in  Chapter  I,  Descrip- 
tive Geometry.     (See  page  36.) 

PLATE  I 

1.  Draw  the  true  direction  of  the  Ray  of  Light  in  a 
1^-inch  Cube. 

2.  Find  graphically  the  true  angle  which  the  Ray  of 
Light  makes  with  either  coordinate. 

Determine  the  shadows  of  the  following  points: 

3.  One  inch  above  the  H  Plane  and  Ij  inches  in  front 
of  the  V  Plane. 

4.  Two  inches  above  the  H  Plane  and  1  inch  in  front 
of  the  V  Plane. 

5.  One  inch  above  the  H  Plane  and  2  inches  in  front 
of  the  V  Plane. 

6.  One  and  one-half  inches  above  the  H  Plane  and  If 
inches  in  front  of  the  V  Plane. 

Determine  the  Visible  and  Imaginary  Shadows  on  the 
coordinates  of  the  following  points : 

7.  One  and  one-half  inches  above  the  H  Plane  and 
1  inch  in  front  of  the  V  Plane. 

8.  One  and  one-quarter  inches  above  the  H  Plane  and 
If  inches  in  front  of  the  V  Plane. 

9.  Lying  in  the  H  Plane  and  1  inch  in  front  of  the  V 
Plane. 

10.  Lying  in  the  7  Plane  and  \\  inches  above  the  H 
Plane. 


64  SHADOWS  OF   POINTS 


PLATE  II 


Making  use  of  the  Profile  and  Horizontal  Projections, 
determine  the  shadows  of  the  following  points : 

1.  One  inch  from  H  and  |  inch  from  V. 

2.  One  inch  from  H  and  1|  inches  from  V. 

3.  One  and  one-quarter  inches  from  H  and  1|  inches 
from  V. 

Making  use  of  the  Profile  and  Vertical  Projections, 
determine  the  shadows  of  the  following  points : 

4.  One  and  one-eighth  inches  from  H  and  f  inch  from  V. 

5.  Three-quarter  inch  from  H  and  1|  inches  from  V. 

6.  Find  the  shadow  of  a  point  on  any  line  shown  by 
its  H  and  V  Projections. 

7.  Find  the  shadow  of  a  point  on  any  oblique  plane. 

8.  Find  the  shadow  of  a  point  1  inch  from  H  and  V, 
on  a. plane  which  is  §  inch  to  its  right  and  perpendicular 
to  V  and  H. 

9.  Find  the  shadow  of  a  point  on  a  plane  whose  traces 
make  an  angle  of  45°  with  both  coordinates,  when  the 
point  is  1  inch  from  H,  1^  inches  from  V,  and  f  inch  from 
the  oblique  plane. 

10.  Find  the  shadow  of  a  point  lying  in  the  Ground- 
Line  paying  special  attention  to  the  notation. 


QUESTIONS   ON   SHADES   AND   SHADOWS  65 

QUESTIONS    ON    SHADES   AND    SHADOWS 

1.  What  causes  objects  to  be  visible?  Of  what  impor- 
tance is  the  Study  of  Shades  and  Shadows  in  connection 
with  architectural  drawings? 

2.  What  and  where  is  the  assumed  source  of  light? 

3.  \Miat  is  the  conventional  direction  for  the  Rays  of 
Light,  and  why  so  determined? 

4.  What  angle  does  the  conventional  Ray  of  Light 
make  with  the  coordinates?  What  angle  do  the  pro- 
jections of  the  Rays  of  Light  make  with  the  coordinates? 

5.  Describe  the  difference  between  a  Ray  of  Light  and 
the  projections  of  a  Ray  of  Light. 

6.  \\Tiat  is  a  Plane  of  Light? 

7.  Into  what  kinds  of  Light  can  Light  be  divided? 
Define  each  kind. 

8.  What  is  Shade  on  an  object? 

9.  WTiat  is  Shadow  on  an  object? 

10.  Define  Shade-Line,  Umbra,  Real  Shades  and  Shadows, 
Imaginary  Shades  and  Shadows,  and  Invisible  Shades  and 
Shadows. 

IL  How  is  the  Shadow  of  any  object  determined? 

12.  Into  what  four  subdivisions  may  the  subject  of 
Shades  and  Shadows  of  objects  be  conveniently  divided? 

13.  Write  all  the  notation  used  in  discussing  the  problems 
of  Shades  and  Shadows. 

14.  Explain  how  the  shadow  of  any  given  point  in  space 
on  any  surface  or  plane  may  be  determined. 

15.  Name  the  different  cases  for  the  finding  of  Shadows 
of  Points  on  Planes. 

16.  State  Axiom  I. 


66  SHADOWS   OF  LINES 


III.     SHADOWS  OF  LINES 


24.  To  find  the  shadow  of  a  given  straight  line  on  a  given 
surface  or  plane. 

Let  ab  be  any  straight  line  in  space.     (Figs.  20  to  26.) 

A  straight  line  is  composed  of  an  infinite  number  of 
points.  If,  therefore,  the  shadows  of  enough  points  are 
located,  the  shadow  of  the  line  can  easily  be  determined. 
Generally  it  is  only  necessary  to  cast  the  shadows  of  the 
points  at  the  extremities  of  the  line,  or  of  the  points  where 
the  shadow  of  the  line  changes^direction.  If  these  shadows 
are  connected  by  straight  lines,  these  lines  will  be  the 
required  shadow-lines.     (Figs.  20  to  26,  Art.  23.) 

The  shadow  of  any  straight  line,  also,  may  be  obtained 
by  finding  the  intersection  of  a  plane  of  light,  containing 
the  given  line,  with  the  surface  receiving  the  shadow.  This 
intersection  contains  the  required  shadow.  (Figs.  21 
and  22.) 

Case  I.  When  the  shadow  of  a  straight  line  falls  on 
the  Vertical  Coordinate  or  elevation. 

Let  a'b"  and  a''b''  be  the  projections  of  a  line  when  its 
shadow  falls  on  the  V  Plane.  Rays  of  Light,  shown  by 
their  projections  Ri"  and  Ri^,  and  Rz"  and  R2',  are  passed 
through  the  points  at  the  extremities  a  and  h  of  the  line. 
These  Rays  of  Light  determine  the  shadows  of  the  two 
points,  a  and  b.  Like  projections  of  the  points  are  con- 
nected, and  this  determines  the  projections  of  the  shadow 
of  the  line.     (Fig.  20.) 

Case  II.  When  the  shadow  of  a  straight  line  falls  on 
the  Horizontal  Coordinate  or  plan. 

This  case  is  the  same  as  Case  I,  except  that  the  shadow 
falls  on  the  //  Plane  instead  of  the  V  Plane.     (Fig.  23.) 

Case  III.  When  the  shadow  of  any  straight  line  falls 


PHIXCIPI.KS    OF    SHADES    AM)    SHADOWS  07 


1  1      II 

Mu  \A 

Fig.  20 


^/^    1 

G     j  I 


Fig.  22 


68 


SHADOWS   OF  LINES 


Fig.  24 


p. 

\ 

\ 

6"^ 

Fig.  26 


Section 


Elevation 


PRINCIPLES   OF   SHADES   AND   SHADOWS  ()9 

on  both  coordinates,  or  on  the  plan  and  elevation,  or  on 
any  oblique  plane. 

Let  ab  be  the  given  straight  line  whose  shadow  falls 
on  both  the  coordinates.     (Figs.  24  and  22.) 

Imaginary  Shadows  are  employed  in  this  Case. 

The  shadow  of  the  entire  line  is  determined  on  the  H 
Plane,  in  front  of  and  behind  the  V  Plane.  (Art.  23, 
Case  IV.)  That  portion  of  the  shadow  behind  V  is  imag- 
inary. The  shadow  of  the  entire  line  is  then  determined 
on  •  the  V  Plane,  above  and  below  the  H  Plane.  That 
portion  of  the  shadow  below  the  H  Plane  is  imaginary. 
The  shadows  on  the  V  and  H  Planes  meet  in  the  GL. 

The  shadow  of  a  straight  line  which_  falls  on  more  than 
one  plane  surface  may;^sq  be  found  by  passing  a  plane 
of  light'TTirough'the  given  line  and  determining  where  this 
plane  intersects  the  plane  surfaces  receiving  the  shadow; 
and  then  passing  Rays  of  Light  through  the  extremities 
of  the  line  to  find  where  the  shadow  on  the  line  of  inter- 
section between  the  plane  of  light  and  the  other  given  planes 
terminates  the  shadow  of  the  line.     (Fig.  22.) 

Case  IV.  To  find  the  shadow  of  a  straight  line  per- 
pendicular to  a  Coordinate  Plane  (either  elevation  or 
plan),  across  several  adjoining  surfaces  r.nd  planes  or  across 
a  series  of  moldings  on  that  coordinate.     (Figs.  25  and  26.) 

Let  ab  and  cd  be  straight  lines,  perpendicular  to  the 
H  and  V  Planes  respectively,  upon  which  is  a  series  of 
moldings. 

A  plane  of  light  is  passed  through  the  lines  and  the 
intersection  of  this  plane  with  the  moldings  dct(>rmined. 
Since  the  shadow  of  the  line  is  contained  in  the  line  of  inter- 
section of  the  plane  of  light  and  the  series  of  moldings,  it 
is  (Fig.  25),  in  Vertical  Projection,  a  line  making  an  angle 
of  45°  with  the  GL;    and   (Fig.  26),  in  Horizontal  Pro- 


70 


SHADOWS  OF   LINES 


Roman  Doric  Order. 


PRINCIPLES   OF  SHADES  AND  SHADOWS 


71 


CuKNlCB. 


72  SHADOWS  OF   LINES 

jection,  a  line  making  an  angle  of  45°  with  the  GL.     (Chap. 
I,  Art.  27.) 

Axiom  II.  The  shadow  of  a  straight  line  perpendicular 
'  to  Oliy  (inhe  coordinates,  is  a  line  making  an  angle  of  45° 
with  the  GL,  in  projection  on  that  coordinate,  or  on  any 
series  of  surfaces,  forms,  or  moldings  on  that  coordinate. 

25.  To  find  the  shadow  of  parallel  lines  on  a  given  plane 
to  which  they  are  parallel. 

Let  ah  and  cd  be  parallel  lines  parallel  to  the  plane  on 
M  hich  their  shadows  fall.     (Figs.  27  and  28.) 

The  shadow  of  each  line  is  determined  on  the  given 
plane.     (Art.  24.) 

This  determines  two  lines  of  shadow  on  the  given  plane, 
jDarallel  to  each  other,  and  parallel  to  the  lines  casting  the 
Hnes  of  shadow. 

Axiom  III.  The  shadow  of  any  line  on  a  plane  to  which 
it  is  parallel,  is  a  line  equal  and  parallel  to  the  given  line. 

Axiom  IV.  The  shadows  of  parallel  hues  on  any  plane 
are  parallel. 

26.  To  find  the  shadow  of  any  line,  not  a  straight  line, 
on  a  given  plane. 

Let  ahcd,  etc.,  be  any  line,  not  a  straight  line,  whose 
shadow  falls  on  a  given  plane.     (Figs.  29,  30,  and  31.) 

.The  line  abed,  etc.,  is  made  up  of  an  infinite  number 
of  points.  By  casting  the  shadows  of  a  sufficient  number 
of  these  points,  the  shadow  of  the  curve  is  readily  deter- 
mined.    (Art.  23.) 

Axiom  V.  The  shadow  of  a  line,  straight  or  otherwise, 
is  determined  by  casting  the  shadows  of  adjacent  points 
of  that  line. 


PRINCIPLES   OF  SHADES   AND   SHADOWS 


73 


Fig.  27 


Fig.  29 


c''     cZ'' 


Fig.  30 


74 


SHADOWS  OF   LINES 


PROBLEMS  ON   CASTING   OF  SHADOWS  OF   LINES      75 


PROBLEMS    ON    THE    CASTING    OF    SHADOWS    OF    LINES 

Note. — In  the  succeeding  problems  the  following  abbre- 
viations are  used:  u  for  up;  /  for  forward;  r  for  right; 
d  for  down;  h  for  back;  and  I  for  left. 


PLATE  III 

Determine  the  shadows  of  the  following  lines. 

1.  A  line  whose  H  Projection  is  1  inch  long,  makes  an 
angle  of  30°  with  the  GL,  slopes  ujr,  and  has  its  lower  end 

I   1  inch  from  H  and  2  inches  from  V.     Its  V  Projection 
I  makes  an  angle  of  30°  with  the  GL. 

2.  A  line  whose  H  Projection  makes  an  angle  of  45° 
I  with  the  GL,  is  1^  inches  long,  slopes  ufl,  and  has  its  lower 
j  end  1  inch  from  V  and  2\  inches  from  H.  Its  V  Projection 
:  makes  an  angle  of  30°  with  the  GL. 

3.  A  line  whose  H  Projection  is  2  inches  long,  makes 
:  an  angle  of  45°  with  the  GL,  ubl,  and  has  its  high  end  § 
i  inch  from  H  and  1|  inches  from  V.  Its  V  Projection  makes 
I  an  angle  of  30°  with  the  GL. 

\         4.  A  line  whose  length  is  1|  inches,  has  one  end  2  inches 
■  from  V  and  §  inch  from  H,  and  is  perpendicular  to  V. 

5.  A  line  whose  length  is  2  inches,  has  one  end  j  inch 
,.  from  V  and  2  inches  from  H,  and  is  perpendicular  to  V. 

6.  A  line  whose  length  is  If  inches,  has  one  end  \  inch 
from  V  and  1  inch  from  H,  and  is  perpendicular  to  V. 

7.  A  line  whose  length  is  U  inches,  is  parallel  to  H, 
has  one  end  1§  inches  from  H  and  |  inch  from  V,  and  is 
not  parallel  to  V. 

8.  A  line  whose  length  is  1^  inches,  is  i)aralk'l  to  V, 
has  one  end  ^  inch  from  H  and  2  inches  from  V,  but  is  not 
parallel  to  H. 


76  SHADOWS   OF   LINES 

9.  A  line  whose  length  is  1|  inches  is  perpendicular  to 
V.     Use  only  the  V  and  Profile-Projections. 

10.  A  line  whose  length  is  1|  inches  and  is  perpendicular 
to  H.     Use  only  the  H  and  V  Projections. 

PLATE  IV 

1.  Find  the  shadow  of  a  2-inch  line  which  is  parallel 
to  the  GL,  1  inch  from  V,  and  1^  inches  from  H. 

2.  Find  the  shadow  of  a  l|-inch  line  which  is  parallel 
to  V,  makes  an  angle  of  30°  with  H,  and  slopes  ur.  The 
lower  corner  is  1  inch  from  V  and  H. 

3.  Find  the  shadow  of  a  2-inch  line  which  is  parallel 
to  V,  makes  an  angle  of  30°  with  H,  and  slopes  ur.  The 
lower  corner  is  2|  inches  from  V  and  1  inch  from  H. 

4.  Find  the  shadow  of  a  2-inch  line  which  is  parallel 
to  V,  makes  an  angle  of  30°  with  H,  and  slopes  ur.  The 
lower  corner  is  1§  inches  from  V  and  1  inch  from  H. 

5.  Find  the  shadow  of  a  line  perpendicular  to  the  Profile 
Plane. 

6.  Find  the  shadow  of  a  2-inch  line  which  is  parallel 
to  the  Profile  Plane,  makes  an  angle  of  45°  with  H  and  V, 
and  has  its  ends  touching  both  H  and  V. 

7.  Find  the  shadow  of  a  line  lying  in  the  Profile  Plane. " 

8.  Find  the  shadow  of  a  2-inch  line  whose  ends  lie  in 
the  H  and  V  Planes,  and  whose  projections  are  perpen- 
dicular to  the  GL. 

PLATE  V 

L  Find  the  shadow  of  two  lines  which  intersect  at  a 
point  2  inches  from  V  and  1^  inches  from  H. 

2.  Find  the  shadow  of  any  two  intersecting  lines  which 
lie  in  an  Imaginary  Plane  perpendicular  to  the  coordinates. 

3.  Find  the  shadow  of  a  curved  line,   four  points  of 


PROBLEMS  OX   CASTING   OF  SHADOWS   OF   LINES        77 

which  are  as  follows:  a,  1  inch  from  H  and  V,  and  |  inch 
from  P;  ?>,  Ij  inches  from  H,  1  inch  from  V,  and  1^  inches 
from  P;  c,  1|  inches  from  H,  11  inches  from  V,  and  2^ 
inches  from  P;  (i,  2  inches  from  /f,  Ij  inches  from  V,  and 
3  inches  from  P. 

4.  Find  the  shadow  of  the  same  line  when  it  is  moved 
1  inch  farther  from  V. 

5.  Find  the  shadow  of  the  same  line  when  it  is  moved 
^  inch  farther  from  V. 

6.  Find  the  shadow  of  a  2-inch  line  which  is  perpen- 
dicular to  the  GL,  and  which  makes  an  angle  of  30°  with 
the  H  Plane.     Its  lower  end  is  ^  inch  from  the  GL. 

7.  Find  the  shadow  of  a  line  and  any  point  in  that 
line. 

QUESTIONS    ON    THE    SHADOWS    OF    LINES 

17.  How  is  the  shadow  of  any  straight  line  determined? 
Explain  both  methods. 

18.  What  is  the  Imaginary  Shadow  of  an  object?  How  is 
it  determined? 

19.  How  is  the  shadow  of  any  line  which  is  not  straight 
determined? 

20.  If  a  straight  line  is  perpendicular  to  one  of  the 
Coordinate  Planes,  what  is  the  character  of  its  shadow  on 
that  plane,  and  on  any  series  of  surfaces,  forms,  or  mold- 
ings on  that  plane? 

21.  What  is  the  character  of  th(>  shadows  of  parallel 
lines  cast  on  a  plane? 

22.  Describe  the  shadow  of  a  lino  on  u  plane  to  which 
it  is  parallel  and  on  which  its  shadow  falls. 


78  SHADOWS  OF  PLANES 

IV.     SHADOWS   OF  PLANES 

27.  To  find  the  shadow  of  any  plane  surface  on  another 
plane  surface.     (Figs.  32  to  35.) 

Let  ahc  be  any  plane  surface. 

Plane  surfaces  are  bounded  by  straight  or  curved  lines. 
Rays  of  Light,  therefore,  are  passed  through  the  points 
at  the  extremities  of  the  straight  lines  or  through  a  suf- 
ficient number  of  points  of  the  curved  lines  to  determine 
the  curve.  The  intersections  of  these  Rays  with  the  sur- 
faces receiving  the  shadow  are  points  of  shadow  in  the 
bounding  line  of  the  shadow.  These  points  are  connected 
by  straight  and  curved  lines  as  required,  and  the  figure 
thus  bounded  is  the  shadow  of  the  plane  surface.  The 
shadows  of  plane  surfaces  may  be  classified,  according  to 
the  location  of  the  shadow,  as  follows : 

Case  I.  When  the  shadow  of  the  plane  surface  falls  on 
the  H  Plane,  or  plan.     (Fig.  32.) 

Case  II.  When  the  shadow  of  the  plane  surface  falls 
on  the  V  Plane,  or  elevation.     (Fig.  33.) 

Case  III.  When  the  shadow  of  the  plane  surface  falls 
on  both  the  H  and  V  planes;  or  on  the  plan  and  elevation. 
(Fig.  34.) 

Case  IV.  When  the  plane  surface  is  perpendicular  to 
one  of  the  coordinates  and  the  shadow  falls  on  that  coordi- 
nnate.     (Fig.  35.) 

Case  V.  When  the  shadow  of  any  plane  surface  falls 
on  any  oblique  plane.     (Fig.  36.) 

28.  The  shadow  of  a  plane  surface  on  another  plane 
surface  to  which  it  is  parallel,  is  equal  in  size  and  shape 
to  the  plane  surface  casting  the  shadow. 

Let  abed  be  a  plane  surface  casting  a  shadow  on  a  parallel 
plane  surface.     (Figs.  37  to  39.) 


PRINCIPLES   OF  SHADES   AND   SHADOWS 


79 


Fig.  33 


ir         c"di 


Fig.  36 


Fig.  35 


80 


SHADOWS  OF  PLANES 


o"        6"  C       d" 


Fig.  37 


Fig.  39 


QUESTIONS   ON    THE   SHADOWS   OF    TLAMIS  81 

Determine  the  shadows  of  the  plane  surfaces  (Art.  27) 
and  it  will  be  seen  that  the  shadows  have  the  size  and  shaj^e 
of  the  plane  surfaces  casting  them. 

Axiom  VI.  Any  point  or  line  lying  on  a  plane  is  the 
Invisible  Shadow  of  itself  on  that  plane. 

29.  The  shadow  of  a  circular  surface  is  determined  by 
circumscribing  a  polygon  about  it.  The  shadow  of  the 
polygon  is  then  determined  and  also  the  points  of  tangency 
of  the  polygon  and  the  circle.  These  points,  connected 
by  a  curve,  determine  the  shadow  of  the  circular  surface 
or  plane.     (Fig.  40.) 

QUESTIONS    ON   THE    SHADOWS   OF    PLANES 

23.  How  is  the  shadow  of  any  plane  surface  determined 
on  another  plane  surface  on  which  it  is  cast? 

24.  How  does  the  shadow  of  any  plane  surface  show- 
on  another  plane  surface  to  which  it  is  parallel? 

25.  How  is  the  shadow  of  a  circular  plane  surface  deter- 
mined on  an  oblique  plane? 


uljiTifil.wijiiUiii 
\ 


A  Section  throuoh  a  RriLniNr,. 


82  SHADES  AND   SHADOWS   OF  SOLIDS 

V.     SHADES  AND  SHADOWS   OF  SOLIDS 

30.  To  find  the  shadow  of  a  Polyhedron  of  such  shape 
that  none  of  its  faces  is  perpendicular  or  parallel  to  the 
coordinates  or  planes  receiving  its  shadow. 

Let  abed  be  any  Polyhedron  with  none  of  its  faces  per- 
pendicular or  parallel  to  the  coordinates.     (Figs.  41  and  42.) 

Polyhedrons  are  bounded  by  plane  surfaces.  The 
shadows,  therefore,  of  the  enclosing  plane  surfaces  are  first 
determined.  These  shadows  together  have  the  form  of  an 
enclosing  polygon  which  is  the  shadow  of  the  polyhedron. 
(Art.  27.) 

The  sides  of  the  enclosing  polygon  of  shadow  are  the 
shadows  of  the  shade-lines  of  the  object  casting  the  shadow. 
Hence  the  shade-lines  are  readily  determined  by  finding 
what  lines  cast  the  bounding  lines  of  the  shadow. 

In  this  figure,  it  is  at  the  outset  impossible  to  determine 
the  shade-line  from  which  to  determine  the  shadow.  Many 
of  "the  Shades  and  Shadows  in  architectural  drawings  have 
to  be  determined  by  the  methods  used  in  this  problem, 
namely,  by  first  determining  the  shadow  of  the  object,  and 
then,  from  the  bounding  shadow-line,  determine  the  shade- 
line  of  the  object. 

31.  To  find  the  shadow  of  a  solid  which  has  its  faces  par- 
allel with  or  perpendicular  to  the  planes  receiving  the  shadow. 

Let  abcdefgh  be  a  solid,  whose  bounding  planes  are 
parallel  or  perpendicular  to  the  planes  receiving  the  shadows. 
(Figs.  43  and  44.) 

This  is  a  problem  which  is  quite  common  in  architec- 
tural drawings,  and  in  which  a  sufficient  number  of  planes 
of  an  object  are  perpendicular  or  parallel  to  the  planes 
receiving  the  shadow,  thus  making  it  possible  to  determine 
the  shadows  by  simple  and  direct  methods.  (Art.  27, 
Case  IV,  and  Art.  28.) 


PRINCIPLES  OF  SHADES   AXD  SHADOWS 


83 


84 


SHADES   AND   SHADOWS   OF   SOLIDS 


/" 


a^b'^ 


■•''d" 


Fig.  43 


PRINCIPLES   f)F   SHADES    AM)   SHADOWS  85 


i^^T 


Fig.  40  «  Fig.  40'' 


Fig.  40*1 


SHADES   AND   SHADOWS  OF  SOLIDS 


Fig.  40« 


Fig.  40* 


PRINCIPLES   OF   SHADES   AND   SHADOWS  87 


Fig.  40  e 


Fig.  40  k 


Fig.  401 


SHADES   AND   SHADOWS   OF  SOLIDS 


PRINCIPLES   OF   .SHADES   AND   SHADOWS  89 

In  this  problem,  the  shadow  is  found  by  first  finding 
the  shade-Une  of  the  soUd  and  then  the  shadow  of  its  shade- 
line.     This  determines  the  bounding  line  of  the  shadow. 

The  shade-line  of  a  simple  object  is  determined  by  find- 
ing where  the  Rays  of  Light  and  the  Planes  of  Light  are 
tangent  to  it. 

The  following  are  some  of  the  problems  frequently  pre- 
sented for  solution : 

Case        I.     A  Prism  with  its  faces  parallel  or  perpen 
dicular    to     the    planes    receiving    the 
shadows.     (Fig.  40a.) 
A  Plinth  on  a  Prism.     (Figs.  406  to  40^.) 
A  Pedestal.     (Fig.  40c.) 
A  Chimney.     (Fig.  40e.) 
A  Dormer.     (Fig.  40/.) 
A  Hand-Rail  on  a  Flight  of  Steps.  (Fig.  47.) 
A  Cylinder.     (Figs.  40m,  40o,  and  40p.) 
A  Cone.     (Figs.  40d,  40k,  401,  and  40m.) 
A  Plinth  on  a  Cylinder.     (Fig.  40r.) 
The  Trim  of  a  Window.     (Fig.  40s.) 

QUESTIONS    ON    THE    SHADES    AND    SHADOWS    OF    SOLIDS 

26.  How  is  the  shadow  of  any  solid  determined? 

27.  Explain  difference  of  methods  of  determining  the 
Shade  and  Shadow  of  a  Polyhedron,  and  a  solid  which  has 
its  bounding  faces  perpendicular  or  parallel  to  the  planes 
receiving  the  shadow. 

28.  Explain  how  the  principle  of  determining  the  shadow 
of  any  point  on  any  plane  enters  into  the  determination 
of  the  shadow  of  any  solid. 

29.  How  is  the  shade-line  of  a  polyhedron  dotormined? 
How  are  the  shade-lines  of  ordinary  simple  objects  deter- 
mined? 


Case 

IL 

Case 

in. 

Case 

IV. 

Case 

V. 

Case 

VI. 

Case 

VII. 

Case  VIIL 

Case 

IX. 

Case 

X. 

90  GENERAL   METHODS 

VI.     GENERAL  METHODS 

32.  There  are  four  principal  methods,  of  which  one  or 
more  are  generally  employed  in  determining  the  Shades 
and  Shadows  of  Architectural  Forms.     They  are : 

(1)  The  Method  of  Oblique  Projection. 

(2)  The  Method  of  Circumscribing  Lines. 

(3)  The  Method  of  Circumscribing  Surfaces. 

(4)  The  Method  of  Slicing. 

33.  The  Method  of  Oblique  Projection  consists  in  passing 
Rays  of  Light  or  Planes  of  Light,  tangent  to  a  given  object, 
to  determine  its  shade-line,  and  in  finding  where  these  Rays 
of  Light  or  Planes  of  Light  intersect  any  other  object  on 
which  the  shadow  of  the  given  object  falls.  (Figs.  45, 
46,  and  47.)  This  method  has  been  the  one  so  far  employed 
in  determining  the  Shades  and  Shadows  of  various  objects, 
and  needs  no  further  explanation. 

34.  The  Method  of  Circumscribing  Lines  consists  in 
drawing  upon  any  solid,  preferably  upon  any  surface  of 
revolution,  a  series  of  lines,  straight  or  curved  as  the  case 
may  require,  which  are,  if  possible,  parallel  to  the  plane 
receiving  the  shadow.  The  required  shadow  of  the  surface 
includes  the  shadows  of  all  the  lines  on  that  surface.  The 
bounding  line  of  the  shadow  is  tangent  to  the  shadow  of 
those  lines  which  cross  the  shade  of  the  surface,  at  points 
which  are  the  shadows  of  the  points  of  crossing.  The  point 
of  intersection  of  two  shadow-lines  is  the  shadow  of  the 
point  of  intersection  of  those  lines  casting  the  shadows, 
if  the  lines  intersect  on  the  surface;  or  the  shadow  of  the 
point  in  which  the  shadow  of  one  line  crosses  the  other, 
when  they  do  not  intersect. 

It  is  evident  from  this  that  if  the  intersection  of  the 
shadows  of  two  lines  is  considered,  that  is,  the  point  of 


PRINCIPLES   OF   SHADES   AM)   SHADOWS 


91 


uW' 


:Kl" 


fV 


chfh 


gMi' 


Fig.  45 


Fig.  46 


a«b"a>^ 


\                                          c'-d'- 

v 

>N                       ro'- 

N. 

\                             /<''A'' 

\^ 

\                              /•• 

\,.r3 

1 

;                   ;  1  1  1 

a'' 

1 
1 

1                             o>' 
1                          rh,l>' 

i     1     1^- 

1                          ,./,,-/, 

1    ,jn. 

X 

1                   a"h'' 

«:-. 

1                   /.''/'' 

t 

y 

/ 

>■ 

V' 

Fig.  47 


92 


GENERAL  METHODS 


PRINCIPLES   OF  SHADES   AND   SHADOWS  93 

tangency  of  the. shadow  of  one  line  with  the  hounding  Hne 
of  the  shadow,  the  points  in  the  shade-hne  on  the  surface 
of  the  object  which  casts  these  points  of  shadow  may  be 
determined  by  passing  back  along  each  Ray  of  Light 
thi'ough  the  point  of  tangency.  A  curved  hne  connecting 
these  points  is  the  shade-hne. 

Let  a  Sphere  (Fig.  49)  be  a  sohd,  whose  surface  is  a 
surface  of  revolut  on,  which  casts  its  shadow  on  the  obhque 
plane  P,  or  on  one  of  the  coordinates.     (Fig.  4S.) 

In  each  case  a  series  of  lines  is  drawn  on  the  surface 
of  the  Sphere  parallel  to  the  plane  receiving  the  shadow  in 
Fig.  48  and  not  parallel  in  Fig.  49.  These  lines  may  be 
determined  by  passing  planes  through  the  Sphere,  parallel 
to  the  given  plane.  The  lines  of  intersection  of  the  assumed 
planes  with  the  Sphere  are  the  lines  required.  The  shadows 
of  these  lines  may  now  be  cast  on  a  plane  to  which  they  are 
parallel.  (Arts.  25  and  28.)  These  shadows  are  now  cir- 
cumscribed bj^  a  curved  line,  which  is  the  bounding  line 
of  the  required  shadow. 

The  shade-line  is  found  by  passing  back  along  the  Rays 
of  Light,  Ri,  Ro,  etc.,  through  the  points  of  tangency  of 
the  bounding  line  of  the  shadow  with  the  shadows  of  the 
assumed  lines  1,  2,  3,  etc.,  and  determining  where  these 
rays  of  light,  Ri,  Ro,  etc.,  are  tangent  to  the  surface  of  the 
Sphere.  These  points,  which  are  points  in  the  shade-line, 
are  connected  b}'  a  curved  line,  which  is  the  reciuired  shade- 
line. 

The  procedure  in  finding  the  Shades  and  Shadows  in 
Fig.  49,  is  the  same,  except  that  the  circumscribed  lines 
are  not  parallel  to  the  surface  receiving  the  shadow. 

35.  The  Method  of  Circumscribing  Surfaces  consists  in 
circumscribing  the  surface  casting  the  shadow,  witli  the 
surface  of  an  object  whose  shade-line  is  readily  determined. 


94 


GENERAL   METHODS 


Tonic  Cap. 


PRINCIPLES   OF  SHADES  AND  SHADOWS  95 


Corinthian  Cap  and  Base. 


96  .  GENERAL   METHODS 

Then  at  a  point  of  tangency  of  the  two  surfaces,  whatever 
is  true  of  one  surface  is  also  true  of  the  other  surface,  for 
such  a  point  is  in  common. 

This  method  is  only  used  in  determining  the  shade-line 
on  double-curved  surfaces  of  revolution. 

Let  it  be  required  to  find  the  shade-line  on  a  Sphere. 
(Fig.  50.) 

The  shade-line  of  a  Cone  is  readily  determined.  The 
Sphere  is  circumscribed  with  a  Cone,  or  with  Cones  in 
different  positions.  The  shade-points  on  the  lines  of  shade 
of  the  Cone  where  the  Cone  is  tangent  to  the  Sphere,  are 
points  of  shade  on  the  shade-line  of  the  Sphere.  These 
points  are  connected  by  a  curved  line,  which  is  the  shade- 
line  required. 

36.  The  Method  of  Slicing  consists  in  cutting  through 
the  object  casting  the  shadow  and  the  object  receiving  the 
shadow  with  planes  of  light  perpendicular  to  one  of  the 
coordinates,  and  in  determining  points  of  Shade  and  Shadow 
by  passing  rays  of  light  through  the  shade-points  in  the 
slices  casting  the  shadow,  and  determining  where  they 
intersect  the  slices  receiving  the  shadow.  (Figs.  51,  52, 
and  53.)  By  the  use  of  this  method  the  shadow  of  any 
object  can  be  found.  It  is,  however,  often  difficult  of  appli- 
cation, as  the  process  of  constructing  the  slices  is  relatively 
slow  and  often  complicated. 

Let  it  be  required  to  find  the  Shades  and  Shadows  of 
a  Sphere,  Torus,  and  Scotia  by  the  Slicing  Method.  (Figs. 
51,  52,  and  53.) 

Planes  of  light  are  passed  through  each  object,  per- 
pendicular to  the  H  Coordinate,  SET,  etc.  The  slices  cut 
out  from  the  object  by  these  planes  are  first  determined. 
(Chap.  I,  Art.  4L)  The  rays  of  light,  Ri  R2,  -etc.,  are 
passed  through  the  points  of  shade  or  the  projecting  por- 


PRINCIPLES  OF  SHADES   AND   SHADOW; 


97 


PRINCIPLES  OF  SHADES   AND   SHADOWS  99 

tions  of  the  slices,  and  the  points  where  these  rays  intersect 
the  sUces  receiving  the  shadow  are  determined.  These 
points  are  connected  by  a  curved  Hne  which  is  the  ro(iiiired 
bounding  Hne  of  the  shadow. 

37.  The  SUcing  Method  is  used  to  great  advantage  in 
determining  which  faces  of  a  Polyhedron  are  in  light  and 
which  in  shade,  since  it  is  sometimes  not  possible  to  deter- 
mine this  by  the  method  of  Oblique  Projection.  The  sides 
in  shade  are  determined  by  passing  a  plane  of  light,  per- 
pendicular to  one  coordinate,  through  the  polyhedron  (Fig. 
54),  and  drawing  the  slice  cut  out  by  this  plane  of  light. 
The  surface  in  shade  are  then  readily  determined  by  passing 
rays  of  light,  and  they  are  the  planes  containing  the  bound- 
ing lines  of  the  slice  which  do  not  receive  the  Rays  of  Light. 

38.  Another  method  of  slicing,  called  the  Slicing  Method 
by  Auxiliary  Planes,  is  sometimes  used.  In  this  method 
the  object  receiving  the  shadow  is  cut  by  Auxiliary  Pianos, 
parallel  to  either  coordinate.  A  portion  of  the  shadow  of 
the  object  is  cast  on  each  Auxiliary  Plane.  The  point  of 
intersection  of  the  shadow-line  on  the  Auxiliary  Plane  with 
the  line  of  intersection  of  the  given  surface  with  the  Auxili- 
ary Plane,  (which  is  a  point  in  common  on  both  planes), 
is  the  shadow  of  a  point  of  the  object  casting  the  shadow 
on  the  object  receiving  the  shadow.  Hence  it  is  a  i)oint 
m  the  shadow-line. 

A  sufficient  number  of  points  of  shadow  may  be  deter- 
mined in  this  way;  and  if  they  are  connected  by  a  line, 
straight  or  curved  as  required,  they  determine  the  nHjuired 
line  of  shadow. 

Let  a  Niche  be  as  shown  (Fig.  5o),  and  determine  its 
shadow  by  this  method. 

Auxiliary  Planes,  RST,  etc.,  are  passed  through  the  Xichc, 
parallel  to  the  V  Plane.     The  shadow  of  the  shade-line  of  the 


100 


GENERAL   METHODS 


Fig.  54 


Fig.  55 


/o««  /' 


/- 


■S'  .^, 


Shadow  of   ah  on  pianes   R-S-T-V 


Plan 
Planes  of  light  90°  with  H  plane 


Shadow  of   ab  en- 
planes  R-T-S-V 


Fig.  56 


Section  on  line^  A  A 


102  GENERAL   METHODS 

niche  is  cast  on  each  of  these  planes  by  casting,  the  shadow  of 
the  center  of  the  Shade4ine  on  the  AuxiUary  Planes,  and 
the  points  in  which  these  shadows  intersect  the  shces  (cut 
from  the  Niche  by  the  Auxihary  Planes),  will  be  points 
a,  b,  c,  d,  e,f,  shown  by  their  projections  a^%  b^%  c^\  d!"',  etc., 
and  a''%  h'%  c",  d'%  etc.,  in  the  required  shadow-line.  These 
points  are  connected  by  a  curve  line,  which  is  the  bound- 
ing line  of  the  required  shadow. 

Let  the  Cornice-Moldings  (composed  of  a  Crown-Mold- 
ing or  Cavetto,  and  an  Ogee-Molding)  at  the  top  of  a  Pedi- 
ment be  as  shown.  (Fig.  56.)  It  is  required  to  find  the 
shadows  of  the  moldings  by  this  method. 

Auxiliary  Slicing  Planes,  SRT,  etc.,  are  passed,  parallel 
to  the  V  Plane,  through  the  moldings,  and  the  slices  cut 
out  are  determined.  The  shadows  of  the  shade-line  a  b, 
and  c  d  then  cast,  on  each  Auxiliary  Plane  in  order  to  deter- 
mine w^here  each  shadow  intersects  the  slices  respectively. 
As  these  are  points  in  common,  on  the  surface  of  the  Pedi- 
ment and  the  Auxiliary  Planes,  they  are  points  in  the 
required  shade-line.  These  points  are  connected  by  a 
curved  line,  which  is  the  boundary-line  of  the  required 
shadow. 

39.  To  find  the  highest  and  low^est  points  in  the  shade- 
line  or  shadow-line  of  a  double-curved  surface  of  revolution. 

Let  a  Sphere  and  Scotia' be  given.  (Fig.  57.)  It  is 
required  to  find  the  highest  and  lowest  points  in  the  shade- 
lines  and  shadow-lines. 

This  problem  is  a  special  application  of  the  Slicing 
Method.  A  Slicing  Plane  of  light,  perpendicular  to  either 
coordinate,  is  passed  through  the  axis  and  center  of  each 
object. 

The  slice  cut  out  is  then  determined,  and  the  Rays  of 
Light  applied,   in  order  to  determine  the  points  of  Shade 


PRINCIPLES   OF   SHADES    AND    SlI AHOW: 


103 


.c  and   V  are  the 

highest  and  lowest  points 

of  shade 


Fig.  57 


x'^J-'"  is  the  highest 
point  of  !tliade 


RPINCIPLES  OF  SHADES   AXl)   SHADOWS  105 

and  Shadow.  (Art.  37.)  These  points  are  the  highest  and 
lowest  points  in  the  shade-hne  or  shadow-Hnes  of  the  Sphere 
or  Scotia,  because  the  sUce  cut  from  the  object  by  the  plane 
of  light  is  a  great  circle  on  which  are  the  h  ghest  and  lowest 
points  of  Shade  and  Shadow  on  the  shade-line  and  shadow- 
line. 

40.  The  forms  used  in  architectural  designs  are  such 
that  it  is  very  often  advantageous  to  find  the  Shades  and 
Shadows  of  an  object  by  more  than  one  of  the  methods 
given. 

Let  it  be  required  to  find  the  Shades  and  Shadows  of  a 
Light-Standard.     (Fig.  58.) 

The  shade  on  the  Sphere  may  be  determined  by  cir- 
cumscribing surfaces.  (Art.  35.)  The  Slicing  IVIethod  deter- 
mines the  shadow  on  the  Scotia.  (Art.  36.)  The  shade 
on  the  Cylinder  may  be  determined  by  the  jNIethod  of 
Oblique  Projection.  The  circumscribing  lines  are  used  on 
the  Torus  to  determine  its  shade. 

In  order  to  find,  therefore,  the  Shades  and  Shadows 
of  an  object,  the  problem  must  be  analyzed,  and  those 
methods  used  which  give  the  quickest  and  easiest  results, 
"and  which  do  not  introduce  complicated  methods  of  con- 
struction. 


106 


GENERAL   METHODS 


FOKVM 

•/Avcvsri 


ROME 


Corinthian  Order 


QUESTIONS  107 

QUESTIONS 

1.  Name  the  different  general  methods  used  for  casting 
the  Shades  and  Shadows  of  sohds. 

2.  Explain  the  Slicing  Method. 

3.  What  is  the  Method  of  Oblique  Projection,  used  to 
determine  the  Shades  and  Shadows  of  an  object'.' 

4.  Explain  the  other  methods  sometimes  used  to  deter- 
mine the  Shades  and  Shadows  of  an  object. 

5.  \\'Tiat  method  is  used  to  determine  the  Shades  and 
Shadows  of  complex  architectural  forms?  Explain  th(' 
usual  procedure  followed  in  determining  the  Shades  and 
Shadows  of  such  forms. 


108 


GENERAL   METHODS 


PLATE  VI 

Find  the  Shades  and  Shadows  for  the  following 
Plates  using  complete  notation  on  all  problems 


^        Ifs"    '', 


IV  V 


n 


III 


( > 

- 

i                ll- 

VI 


X 


VII 


vm 


XI 


a 


IX 


^■r^^ 


PROIU.KMS 


109 


no 


GENERAL   METHODS 


PLATE  Vin 


II 


K.I    ■'•  1 


VI 


PROBLEMS 


111 


PLATE  IX 

Find  only  shadows  of  the  lines  in  these  problemi 


Section  or  Profile 


ni 


IV 


112 


GENERAL   METHODS 


PLATE  X 


./      /    / 1 
'         /    / 

y^  t         j  I 

T; 1^  ^^  C\)  /  CO 

I       \    ^^^  /  /  ' 

2  i  V>^/ ! 

:,   i  I  I     I  ^: 

I  I  I     I 

II  I     12 
'         '  !    ' 

M    I         I  I    j-J 

I         I  I'i^^ 
I       J--"^^^    I  / 
I  -^^  "''       I  / 

-^^^--<  I  A, 


I        I         I  I    I 
«    I        '         1  I    ' 


ni 


,t. 

^     ^'^"    . 

s 

'2 

L. 

•J 

v^ 

' — ] 

-J 

- 

p> 

I 

V 

PROBLEMS 


li:"{ 


PLATE  XI 


f  '  'i  y^ 2 ^  ^  'a  ^ 

-V 

~f ' i 

Ix 


— : 

1" 

2" 

-■ 

eg 

-J 

-) 

^ 

L 

L 

L-— -- 

t p^ 

r    "■ 

-L 

:i? 

?i" 

^,' 

'»■ 

,r 

2'                 '^ 

CJ 

'             1 

*;.t 

F- 

'^^     * 

!■■ 

'4--; 

'    ) 

Vi' 


■?f 


•"■I 

— I 


III 


IV 


114 


GENERAL  METHODS 


PLATE  Xn 


Use  Oblique  Projection 


Use  Circumscribing  Lines 


Use 

Slicins 
Method 


III 
Use  Circumscribing  Surfaces 


IV 


PRC^BLEMS 


115 


PLATE  Xm 


Plan 

—  4' 


Find  the  Shades  and  Shadows   by  three  different  methods. 


116 


GENERAL   METHODS 


' 

PLA'l'iii  XIV 

) 

K 

(^ 

1        '^\ 

1 

I? 

t 

!'" 

.     y 

y- 

1  ( 

\ 

1 
\ 

Elevation 

Plan 

PHOBLKMS 


118 


GENERAL   METHODS 


PLATE  XVI 


•^h 


Profile 


T^rr 


.n  r 


2Vi" 


Elevation 


II 


PROBLEMS 


119 


PLATE  XVII 


j^==^ 


Profile 


Elevation 


120 


GENERAL   METHODS 


PLATE  XVIII 


Elevation 


Plan 


Section 


PROBLEMS 


121 


PLATE  XIX 


C*=S(|n 


I 


cW8=3(!Q 


122 


GENERAL   METHODS 


PROBLEMS 


12:^ 


PLATE  XXI 


n 


Elevation 


Section 


These  Problems  are  to  be  done  at  twice  this  scale 


124 


GENERAL   METHODS 


PROBLEMS 


12: 


PLATE  XXIII 


This  Problem  to  be  worked  out  at  twice  this  »culc 


126 


GENERAL   METHODS 


PLATE  XXIV    - 

: 

1", 

4'// 

1\ 

4H" 

<  l"> 

Light 
Wash 

'« 

Medium 
Wash 

- 

Dark 
Wash 

00 

Light  to 
Medium  to 
Dark  Wash 

I 

Date 

- 

Name 

' 

PROHLEMS 


127 


128 


GENERAL  METHODS 


PLATE  XXVI 


/Dark  to\ 
/       Light     \ 

1^     . 

^          2"             ^ 

^c 

Other  washes  and  dimensions  same  as  Plate  II 


..  1 

1 

lU 

Vi" 

,     1>4"  . 

m" 

Light 

tor 

)ark 

'^ 

■ 

\e< 

- 

Medium 

„ 

Light  to  Dark/ 

„ 

/    Light 

/       /^^Dark  to 
/        /        Light 

^-^ 

PRINCIPLES   OF   SIIADKS    AND   SHADOWS 


\2\) 


Fig.  10. 


fmMii^^^i^ 


Fig.  11. 


130  WASH-RENDERING 


VII.     WASH-RENDERING 


41.  An  architect's  drawing  is  a  representation  of  a 
structure  as  it  will  appear  in  reality. 

The  natural  laws  of  light  and  shade  are  followed  in  the 
conventional  manner  explained  in  the  foregoing  pages  of 
this  book  to  the  extent  that  it  will  advance  this  represen- 
tation. One  medium  used  in  the  wash-renderings  of  draw- 
ings is  India  ink,  especially  in  the  presentation  of  drawings 
submitted  in  competitions. 

The  materials  needed  for  wash-rendering  are  Chinese, 
Japanese,  or  India  stick-ink,  an  ink-slab,  a  small  and  a 
large  sable-hair  brush,  a  nest  of  saucers,  a  sponge,  and  a 
bowl  or  pan.  Whatman's  paper  (rough)  is  a  good  paper 
for  wash-drawings. 

To  stretch  the  paper,  the  entire  back  of  the  sheet  is 
wet,  except  a  1-inch  margin  around  the  edge,  which  is  kept 
dry  for  the  paste.  After  the  paste  is  smeared  on  the  edges, 
the  surplus  water  is  removed,  and  the  sheet  turned  over. 
The  right  side,  except  the  edges,  is  then  wet  and  the  edges 
pasted  down  starting  from  the  middle  of  the  sides.  Gently 
stretch  it,  and  work  towards  the  corners,  which  are  pasted 
down  last.  All  the  water  should  then  be  taken  up  before 
leaving  the  sheet  to  dry.  Care  should  be  exercised  to  pre- 
vent the  water  from  wetting  the  edges.  When  they  are 
wet  the  paper  dries  before  the  paste  adheres  to  the  board, 
and  loosens  the  edges.  Boards  which,  when  wet,  stain 
paper  should  not  be  used. 

Soft  pencils  should  not  be  used  in  making  these  draw- 
ings. The  HB,  H,  and  F  ''Kohinoor"  or  similar  grade  of 
pencil  give  excellent  results.  Care  should  be  taken  in 
making  the  drawings  themselves,  as  a  good  pencil-drawing 
with  careless  washes  generally  presents  a  better  final  appear- 


PRINCIPLES   UP   .SHADES   AM)   SHADOWS 


131 


Student  Work— State  College  of  Washington. 


^s^  *=s 


A  Museum. 
Student  Work— Clcinsoii  College. 


WASH-RENDERING 


Hr^-^^^"^^^ 


A  Museum, 
Student  Work. 


n 


/:::._  _n'_ 


A  Museum. 
Student  Work. 


PRINCIPLES  OF  SHADES   AND  SHADOWS  133 

ance  than  a  poor  drawing  with  more  carefully  made  washes. 
Good  washes  never  disguise  poor  drawings.  A  long,  round 
point  (not  a  chisel-point)  should  be  kept  on  the  pencil  at 
all  times.  A  point  of  this  kind  can  be  made  with  a  sand- 
paper sharpener.  When  used  to  draw  a  line,  the  pencil 
should  be  pulled  across  the  sheet  and  twisted  at  the  same 
time,  in  order  to  keep  the  point  sharp.  A  line  should  be 
of  uniform  width,  should  have  a  definite  beginning  and  end 
of  a  firm  touch,  and  should  not  taper  off  and  tlisappear 
gradually.  All  construction-lines  can  run  past  their  inter- 
sections. It  is  not  necessary  to  rub  out  construction-lines, 
but  they  should  be  drawn  very  light. 

For  large  washes  the  stick  India  ink  should  be  freshly 
ground  on  a  clean  slab  each  time  it  is  required.  Ink,  after 
standing  several  hours,  should  be  thrown  away.  The  ink 
is  rubbed  on  a  slab  until  the  solution  is  very  black,  and 
allowed  to  settle  in  the  covered  slab.  Ink-wash  evaporates 
rapidly,  and  dust  settles  in  an  uncovered  slab.  After  the 
ink  has  settled,  enough  for  the  required  wash  is  taken  from 
the  top  with  the  point  of  the  brush  and  put  with  clean 
water  in  a  saucer.  The  ink  should  always  be  taken  from 
the  top,  with  no  disturbance  of  the  sediment,  as  the  latter 
streaks  and  spots  the  wash. 

In  laying  on  a  wash,  plenty  of  the  color-solution  used 
is  kept  on  the  paper  and  in  the  brush,  and  the  wash  is 
''floated"  on.  In  bringing  the  color  up  to  the  boundary-lines, 
it  is  squeezed  from  the  brush  and  brought  to  a  sharp  point. 
The  brush  is  then  used  to  bring  the  wash  to  \ho  l^oundary- 
line.  This  is  done,  also,  as  the  bottom  of  the  wash  is 
approached,  and  all  the  surplus  color  is  then  taken  up  just 
before  reaching  the  bottom.  Where  only  a  small  surface 
is  to  be  rendered,  the  tint  may  be  mixed  on  a  \nrrr  of  jiapcr 
in  the  same  manner  as  in  th(^  saucer. 


134  WASH-RENDERING 

Beginners  have  difficulty  in  keeping  a  wash  from 
''streaking."  Much  skill  is  necessary  to  prevent  this  and 
is  acquired  only  by  practice.  Some  other  cause  of  streaks 
are,  dust  on  the  paper,  grease  on  the  paper  through  contact 
with  the  hands,  water-streaks  or  marks,  and  a  disturbed 
surface  from  the  use  of  a  hard  rubber. 

Before  applying  a  wash  the  drawing  should  be  sponged 
off.  If  a  rubber  is  used,  it  should  be  soft  and  used  before 
this  washing.  A  skillful  handling  of  the  brush  is  acquired 
through  constant  practice.  It  is  held  like  a  pencil,  with 
the  arm  free  and  not  resting  on  the  drawing,  but  with  the 
little  finger  in  contact  with  the  drawing.  Only  the  point 
of  the  brush  should  touch  the  paper.  In  laying  on  the  wash 
the  board  is  slightly  tilted  in  the  direction  in  which  the 
wash  is  being  applied.  The  wash  should  be  applied  so 
that  all  portions  of  the  paper  are  throughly  and  evenly 
wet.  A  portion  partly  dried  should  not  be  again  wet  in 
that  condition,  and  the  draughtsman  should  wait  until  it 
has  thoroughly  dried. 

Washes  which  grade  gradually  are  made  by  the  addition 
of  water  or  color  each  time  there  is  an  advance.  These 
additions  must  be  carefully  managed  so  that  the  change 
in  color  is  even.  It  is  difficult  to  lay  an  evenly  'graded 
wash  in  one  application.  It  is  better  to  first  lay  a  light, 
flat,  or  slightly  graded  wash,  and  then  the  graded  wash, 
in  one  or  several  operations.  The  ''washing  out"  of  a 
wash  should  be  avoided;  but  if  it  is  necessary,  a  sponge 
and  a  great  deal  of  water  should  be  used;  or  the  drawing 
should  be  placed  under  a  faucet,  the  water  turned  on,  and 
the  sponge  used  on  the  parts  to  be  lightened.  Sometimes 
gradations  are  obtained  by  laying  on  successive  flat  washes, 
each  wash  beginning  a  little  lower  in  value  than  the  previous 
cnc,  or  vice  versa. 


PRINCIPLES   OF  SHADES   AND  SHADOWS  135 

In  laying  on  a  wash,  a  good-sized  "puddle"  should  be 
kept  on  the  sheet.  The  bottom  of  the  wash  must  be  kept 
wet  or  it  will  streak  The  tint  should  be  carried  down 
evenly  across  the  board,  and  the  brush  moved  rapidly  from 
side  to  side.  The  center  of  the  wash  is  kept  lower  than 
the  sides.  Before  putting  on  a  wash  it  is  a  good  plan  to 
dampen  the  paper  with  a  soft  sponge  so  as  to  have  the  wash 
applied  more  evenly. 

Washes  which  are  put  on  after  the  space  has  l)een 
sponged  and  before  the  paper  is  entirely  dry,  that  is,  while 
the  paper  is  still  damp  but  not  wet  enough  to  make  the 
wash  run,  usually  give  good  results. 

In  the  following  plates  there  are  three  grades  or  values. 
Medium  is  supposed  to  be  half  way  in  value  between  light 
and  dark.  The  spaces  are  to  be  drawn  in  with  a  soft  pencil, 
on  Whatman's  cold-pressed  paper. 

PLATE  27 

1.  Draw  a  corner  of  Vignola's  Doric  Order,  with  its 
Entablature,  making  the  Column  12  inches  high.  Cast 
all  the  Shades  and  Shadows.  Wash  in  the  Shades  and 
Shadows,  background,  etc.,  with  graded  ink-washes. 

PLATE  28 
1.  Do  the  same  as  in  Plate  IV,  for  the  Ionic  Order. 

PLATE  29 
1.  Do  the  same  as  in  Plate  IV,  for  the  Corinthian  Order. 


Wiley  Special  Subject  Catalogues 

For  convenience  a  list  of  ihe  Wiley  Special  Subject 
Catalogues,  envelope  size,  has  been  printed.  These 
are  arranged  in  groups — each  catalogue  having  a  key 
symbol.  (See  special  Subject  List  Below).  To 
obtain  any  of  these  catalogues,  send  a  postal  using 
the  key  symbols  of  the  Catalogues  desired. 


1 — ^Agriculture.     Animal  Husbandrj'.     Dairying.     Industrial 
Canning  and  Preserving. 

2 — Architecture.       Building.       Concrete  and  Masonry. 

3 — Business  Administration  and  Management.     Law. 

Industrial  Processes:   Canning  and  Preserving;     Oil  and  Ga« 
Production;  Paint;  Printing;  Sugar  Manufacture;  Textile. 

CHEMISTRY 
4a  General;  Analj-tical,  Qualitative  and  Quantitative;  Inorganic; 

Organic. 
4b  Electro-  and  Physical;  Food  and  Water;    Industrial;    Medical 

and  Pharmaceutical;  Sujar. 

CIVIL  ENGINEERING 

5a  Unclassified  and  Structural  Engineering. 

5b  Materials  and  Mechanics  of  Construction,  including;  Cement 
and  Concrete;  Excavation  and  Earthwork;  Foundations; 
Masonry. 

5c    Railroads;  Surveying. 

5d  Dams;  Hydraulic  Engineering;  Pumpin)?  and  Hydraulics;  Irri- 
gation Engineering;  River  and  Harbor  Engineering;  Water 
Supply. 


CIVIL  ENGINEERING— CoK/iwMfi 
5e   Highways;     Municipal     Engineering;     Sanitary     Engineering; 
Water    Supply.      Forestry.      Horticulture,    Botany    and 
Landscape  Gardening. 


6 — ^Design.       Decoration.       Drawing:     General;     Descriptive 

Geometry;  Kinematics;  Mechanical. 

ELECTRICAL  ENGINEERING— PHYSICS 

7 — General  and  Unclassified;  Batteries;  Central  Station  Practice; 
Distribution  and  Transmission;  Dynamo-Electro  Machinery; 
Electro-Chemistry  and  Metallurgy;  Measuring  Instruments 
and  Miscellaneous  Apparatus. 


8 — Astronomy.      Meteorology.      Explosives.      Marine    and 
Naval  Engineering.     Military.     Miscellaneous  Books. 

MATHEMATICS 
9 — General;    Algebra;   Analytic  and   Plane   Geometry;    Calculus; 
Trigonometry;  Vector  Analysis. 

MECHANICAL  ENGINEERING 
10a  General  and  Unclassified;  Foundry  Practice;  Shop  Practice. 
10b  Gas  Power  and    Internal   Combustion   Engines;   Heating  and 

Ventilation;  Refrigeration. 
10c  Machine  Design  and  Mechanism;  Power  Transmission;  Steam 

Power  and  Power  Plants;  Thermodynamics  and  Heat  Power. 
11 — Mechanics.    .  . 

12 — Medicine.  Pharmacy.  Medical  and  Pharmaceutical  Chem- 
istry. Sanitary  Science  and  Engineering.  Bacteriology  and 
Biolog>'. 

MINING  ENGINEERING 

13 — General;  Assaying;  Excavation,  Earthwork,  Tunneling,  Etc.; 
Explosives;  Geology;  Metallurgy;  Mineralogy;  Prospecting; 
Ventilation . 

14 — Food  and  Water.  Sanitation.  Landscape  Gardening. 
Design  and  Decoration.     Housing,  House  Painting. 


„,™kn  lo  dbIk^^m  which  borrowed 

LOAN  DEPT. 

Renewed  books  are  subjea .o  .mmeia«r^ 


General  Ubtary 


*-Mp 


